From 72ef564962f6ac6383671fb78ea25e4d07872304 Mon Sep 17 00:00:00 2001 From: fjosw Date: Thu, 28 May 2026 09:38:10 +0000 Subject: [PATCH] Documentation updated --- docs/pyerrors/input/openQCD.html | 3108 +++++++++++++++--------------- docs/search.js | 2 +- 2 files changed, 1558 insertions(+), 1552 deletions(-) diff --git a/docs/pyerrors/input/openQCD.html b/docs/pyerrors/input/openQCD.html index d08b20d0..25893579 100644 --- a/docs/pyerrors/input/openQCD.html +++ b/docs/pyerrors/input/openQCD.html @@ -337,1077 +337,1081 @@ 238 239 It is assumed that one measurement is performed for each config. 240 If this is not the case, the resulting idl, as well as the handling - 241 of r_start, r_stop and r_step is wrong and the user has to correct + 241 of `r_start`, `r_stop` and `r_step` is wrong and the user has to correct 242 this in the resulting observable. - 243 - 244 Parameters - 245 ---------- - 246 path : str - 247 Path to .ms.dat files - 248 prefix : str - 249 Ensemble prefix - 250 dtr_read : int - 251 Determines how many trajectories should be skipped - 252 when reading the ms.dat files. - 253 Corresponds to dtr_cnfg / dtr_ms in the openQCD input file. - 254 xmin : int - 255 First timeslice where the boundary - 256 effects have sufficiently decayed. - 257 spatial_extent : int - 258 spatial extent of the lattice, required for normalization. - 259 postfix : str - 260 Postfix of measurement file (Default: ms) - 261 r_start : list - 262 list which contains the first config to be read for each replicum. - 263 r_stop : list - 264 list which contains the last config to be read for each replicum. - 265 r_step : int - 266 integer that defines a fixed step size between two measurements (in units of configs) - 267 If not given, r_step=1 is assumed. - 268 plaquette : bool - 269 If true extract the plaquette estimate of t0 instead. - 270 names : list - 271 list of names that is assigned to the data according according - 272 to the order in the file list. Use careful, if you do not provide file names! - 273 files : list - 274 list which contains the filenames to be read. No automatic detection of - 275 files performed if given. - 276 assume_thermalization : bool - 277 If True: If the first record divided by the distance between two measurements is larger than - 278 1, it is assumed that this is due to thermalization and the first measurement belongs - 279 to the first config (default). - 280 If False: The config numbers are assumed to be traj_number // difference - 281 - 282 Returns - 283 ------- - 284 E_dict : dictionary - 285 Dictionary with the flowed action density at flow times t - 286 """ - 287 - 288 if 'files' in kwargs: - 289 known_files = kwargs.get('files') - 290 else: - 291 known_files = [] - 292 - 293 ls = _find_files(path, prefix, postfix, 'dat', known_files=known_files) - 294 - 295 replica = len(ls) - 296 - 297 if 'r_start' in kwargs: - 298 r_start = kwargs.get('r_start') - 299 if len(r_start) != replica: - 300 raise Exception('r_start does not match number of replicas') - 301 r_start = [o if o else None for o in r_start] - 302 else: - 303 r_start = [None] * replica - 304 - 305 if 'r_stop' in kwargs: - 306 r_stop = kwargs.get('r_stop') - 307 if len(r_stop) != replica: - 308 raise Exception('r_stop does not match number of replicas') - 309 else: - 310 r_stop = [None] * replica - 311 - 312 if 'r_step' in kwargs: - 313 r_step = kwargs.get('r_step') - 314 else: - 315 r_step = 1 - 316 - 317 print('Extract flowed Yang-Mills action density from', prefix, ',', replica, 'replica') - 318 - 319 if 'names' in kwargs: - 320 rep_names = kwargs.get('names') - 321 else: - 322 rep_names = [] - 323 for entry in ls: - 324 truncated_entry = entry.split('.')[0] - 325 idx = truncated_entry.index('r') - 326 rep_names.append(truncated_entry[:idx] + '|' + truncated_entry[idx:]) - 327 - 328 Ysum = [] - 329 - 330 configlist = [] - 331 r_start_index = [] - 332 r_stop_index = [] - 333 - 334 for rep in range(replica): - 335 - 336 with open(path + '/' + ls[rep], 'rb') as fp: - 337 t = fp.read(12) - 338 header = struct.unpack('iii', t) - 339 if rep == 0: - 340 dn = header[0] - 341 nn = header[1] - 342 tmax = header[2] - 343 elif dn != header[0] or nn != header[1] or tmax != header[2]: - 344 raise Exception('Replica parameters do not match.') - 345 - 346 t = fp.read(8) - 347 if rep == 0: - 348 eps = struct.unpack('d', t)[0] - 349 print('Step size:', eps, ', Maximal t value:', dn * (nn) * eps) - 350 elif eps != struct.unpack('d', t)[0]: - 351 raise Exception('Values for eps do not match among replica.') - 352 - 353 Ysl = [] - 354 - 355 configlist.append([]) - 356 while True: - 357 t = fp.read(4) - 358 if (len(t) < 4): - 359 break - 360 nc = struct.unpack('i', t)[0] - 361 configlist[-1].append(nc) - 362 - 363 t = fp.read(8 * tmax * (nn + 1)) - 364 if kwargs.get('plaquette'): - 365 if nc % dtr_read == 0: - 366 Ysl.append(struct.unpack('d' * tmax * (nn + 1), t)) - 367 t = fp.read(8 * tmax * (nn + 1)) - 368 if not kwargs.get('plaquette'): - 369 if nc % dtr_read == 0: - 370 Ysl.append(struct.unpack('d' * tmax * (nn + 1), t)) - 371 t = fp.read(8 * tmax * (nn + 1)) - 372 - 373 Ysum.append([]) - 374 for i, item in enumerate(Ysl): - 375 Ysum[-1].append([np.mean(item[current + xmin: - 376 current + tmax - xmin]) - 377 for current in range(0, len(item), tmax)]) - 378 - 379 diffmeas = configlist[-1][-1] - configlist[-1][-2] - 380 configlist[-1] = [item // diffmeas for item in configlist[-1]] - 381 if kwargs.get('assume_thermalization', True) and configlist[-1][0] > 1: - 382 warnings.warn('Assume thermalization and that the first measurement belongs to the first config.') - 383 offset = configlist[-1][0] - 1 - 384 configlist[-1] = [item - offset for item in configlist[-1]] - 385 - 386 if r_start[rep] is None: - 387 r_start_index.append(0) - 388 else: - 389 try: - 390 r_start_index.append(configlist[-1].index(r_start[rep])) - 391 except ValueError: - 392 raise Exception('Config %d not in file with range [%d, %d]' % ( - 393 r_start[rep], configlist[-1][0], configlist[-1][-1])) from None - 394 - 395 if r_stop[rep] is None: - 396 r_stop_index.append(len(configlist[-1]) - 1) - 397 else: - 398 try: - 399 r_stop_index.append(configlist[-1].index(r_stop[rep])) - 400 except ValueError: - 401 raise Exception('Config %d not in file with range [%d, %d]' % ( - 402 r_stop[rep], configlist[-1][0], configlist[-1][-1])) from None - 403 - 404 if np.any([len(np.unique(np.diff(cl))) != 1 for cl in configlist]): - 405 raise Exception('Irregular spaced data in input file!', [len(np.unique(np.diff(cl))) for cl in configlist]) - 406 stepsizes = [list(np.unique(np.diff(cl)))[0] for cl in configlist] - 407 if np.any([step != 1 for step in stepsizes]): - 408 warnings.warn('Stepsize between configurations is greater than one!' + str(stepsizes), RuntimeWarning) - 409 - 410 idl = [range(configlist[rep][r_start_index[rep]], configlist[rep][r_stop_index[rep]] + 1, r_step) for rep in range(replica)] - 411 E_dict = {} - 412 for n in range(nn + 1): - 413 samples = [] - 414 for nrep, rep in enumerate(Ysum): - 415 samples.append([]) - 416 for cnfg in rep: - 417 samples[-1].append(cnfg[n]) - 418 samples[-1] = samples[-1][r_start_index[nrep]:r_stop_index[nrep] + 1][::r_step] - 419 new_obs = Obs(samples, rep_names, idl=idl) - 420 E_dict[n * dn * eps] = new_obs / (spatial_extent ** 3) - 421 - 422 return E_dict - 423 + 243 The function also assumes that `r_step` is the same across all replica. + 244 + 245 Parameters + 246 ---------- + 247 path : str + 248 Path to .ms.dat files + 249 prefix : str + 250 Ensemble prefix + 251 dtr_read : int + 252 Determines how many trajectories should be skipped + 253 when reading the ms.dat files. + 254 Corresponds to dtr_cnfg (dncnfg) in the openQCD input file. + 255 xmin : int + 256 First timeslice where the boundary + 257 effects have sufficiently decayed. + 258 spatial_extent : int + 259 spatial extent of the lattice, required for normalization. + 260 postfix : str + 261 Postfix of measurement file (Default: ms) + 262 r_start : list + 263 list which contains the first config to be read for each replicum. + 264 r_stop : list + 265 list which contains the last config to be read for each replicum. + 266 r_step : int + 267 integer that defines a fixed step size between two measurements (in units of configs) + 268 If not given, r_step=1 is assumed. + 269 plaquette : bool + 270 If true extract the plaquette estimate of t0 instead. + 271 names : list + 272 list of names that is assigned to the data according according + 273 to the order in the file list. Use careful, if you do not provide file names! + 274 files : list + 275 list which contains the filenames to be read. No automatic detection of + 276 files performed if given. + 277 assume_thermalization : bool + 278 If True: If the first record divided by the distance between two measurements is larger than + 279 1, it is assumed that this is due to thermalization and the first measurement belongs + 280 to the first config (default). + 281 If False: The config numbers are assumed to be traj_number // difference + 282 + 283 Returns + 284 ------- + 285 E_dict : dictionary + 286 Dictionary with the flowed action density at flow times t + 287 """ + 288 + 289 if 'files' in kwargs: + 290 known_files = kwargs.get('files') + 291 else: + 292 known_files = [] + 293 + 294 ls = _find_files(path, prefix, postfix, 'dat', known_files=known_files) + 295 + 296 replica = len(ls) + 297 + 298 if 'r_start' in kwargs: + 299 r_start = kwargs.get('r_start') + 300 if len(r_start) != replica: + 301 raise Exception('r_start does not match number of replicas') + 302 r_start = [o if o else None for o in r_start] + 303 else: + 304 r_start = [None] * replica + 305 + 306 if 'r_stop' in kwargs: + 307 r_stop = kwargs.get('r_stop') + 308 if len(r_stop) != replica: + 309 raise Exception('r_stop does not match number of replicas') + 310 else: + 311 r_stop = [None] * replica + 312 + 313 if 'r_step' in kwargs: + 314 r_step = kwargs.get('r_step') + 315 else: + 316 r_step = 1 + 317 + 318 print('Extract flowed Yang-Mills action density from', prefix, ',', replica, 'replica') + 319 + 320 if 'names' in kwargs: + 321 rep_names = kwargs.get('names') + 322 else: + 323 rep_names = [] + 324 for entry in ls: + 325 truncated_entry = entry.split('.')[0] + 326 idx = truncated_entry.index('r') + 327 rep_names.append(truncated_entry[:idx] + '|' + truncated_entry[idx:]) + 328 + 329 Ysum = [] + 330 + 331 configlist = [] + 332 r_start_index = [] + 333 r_stop_index = [] + 334 + 335 for rep in range(replica): + 336 + 337 with open(path + '/' + ls[rep], 'rb') as fp: + 338 t = fp.read(12) + 339 header = struct.unpack('iii', t) + 340 if rep == 0: + 341 dn = header[0] + 342 nn = header[1] + 343 tmax = header[2] + 344 elif dn != header[0] or nn != header[1] or tmax != header[2]: + 345 raise Exception('Replica parameters do not match.') + 346 + 347 t = fp.read(8) + 348 if rep == 0: + 349 eps = struct.unpack('d', t)[0] + 350 print('Step size:', eps, ', Maximal t value:', dn * (nn) * eps) + 351 elif eps != struct.unpack('d', t)[0]: + 352 raise Exception('Values for eps do not match among replica.') + 353 + 354 Ysl = [] + 355 + 356 configlist.append([]) + 357 while True: + 358 t = fp.read(4) + 359 if (len(t) < 4): + 360 break + 361 nc = struct.unpack('i', t)[0] + 362 if nc % dtr_read == 0: + 363 configlist[-1].append(nc) + 364 t = fp.read(8 * tmax * (nn + 1)) + 365 if kwargs.get('plaquette'): + 366 if nc % dtr_read == 0: + 367 Ysl.append(struct.unpack('d' * tmax * (nn + 1), t)) + 368 t = fp.read(8 * tmax * (nn + 1)) + 369 if not kwargs.get('plaquette'): + 370 if nc % dtr_read == 0: + 371 Ysl.append(struct.unpack('d' * tmax * (nn + 1), t)) + 372 t = fp.read(8 * tmax * (nn + 1)) + 373 + 374 Ysum.append([]) + 375 for i, item in enumerate(Ysl): + 376 Ysum[-1].append([np.mean(item[current + xmin: + 377 current + tmax - xmin]) + 378 for current in range(0, len(item), tmax)]) + 379 + 380 diffmeas = configlist[-1][-1] - configlist[-1][-2] + 381 if not all(c % diffmeas == 0 for c in configlist[-1]): + 382 raise ValueError(f"Irregular spacing of configurations in {ls[rep]}, determined stepsize does not divide all trajectory steps.") + 383 configlist[-1] = [item // diffmeas for item in configlist[-1]] + 384 if kwargs.get('assume_thermalization', True) and configlist[-1][0] > 1: + 385 warnings.warn('Assume thermalization and that the first measurement belongs to the first config.') + 386 offset = configlist[-1][0] - 1 + 387 configlist[-1] = [item - offset for item in configlist[-1]] + 388 + 389 if r_start[rep] is None: + 390 r_start_index.append(0) + 391 else: + 392 try: + 393 r_start_index.append(configlist[-1].index(r_start[rep])) + 394 except ValueError: + 395 raise Exception('Config %d not in file with range [%d, %d]' % ( + 396 r_start[rep], configlist[-1][0], configlist[-1][-1])) from None + 397 + 398 if r_stop[rep] is None: + 399 r_stop_index.append(len(configlist[-1]) - 1) + 400 else: + 401 try: + 402 r_stop_index.append(configlist[-1].index(r_stop[rep])) + 403 except ValueError: + 404 raise Exception('Config %d not in file with range [%d, %d]' % ( + 405 r_stop[rep], configlist[-1][0], configlist[-1][-1])) from None + 406 + 407 if np.any([len(np.unique(np.diff(cl))) != 1 for cl in configlist]): + 408 raise Exception('Irregular spaced data in input file!', [len(np.unique(np.diff(cl))) for cl in configlist]) + 409 stepsizes = [list(np.unique(np.diff(cl)))[0] for cl in configlist] + 410 if np.any([step != 1 for step in stepsizes]): + 411 warnings.warn('Stepsize between configurations is greater than one!' + str(stepsizes), RuntimeWarning) + 412 + 413 idl = [range(configlist[rep][r_start_index[rep]], configlist[rep][r_stop_index[rep]] + 1, r_step) for rep in range(replica)] + 414 E_dict = {} + 415 for n in range(nn + 1): + 416 samples = [] + 417 for nrep, rep in enumerate(Ysum): + 418 samples.append([]) + 419 for cnfg in rep: + 420 samples[-1].append(cnfg[n]) + 421 samples[-1] = samples[-1][r_start_index[nrep]:r_stop_index[nrep] + 1][::r_step] + 422 new_obs = Obs(samples, rep_names, idl=idl) + 423 E_dict[n * dn * eps] = new_obs / (spatial_extent ** 3) 424 - 425def extract_t0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs): - 426 """Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs. + 425 return E_dict + 426 427 - 428 It is assumed that all boundary effects have - 429 sufficiently decayed at x0=xmin. - 430 The data around the zero crossing of t^2<E> - c (where c=0.3 by default) - 431 is fitted with a linear function - 432 from which the exact root is extracted. - 433 - 434 It is assumed that one measurement is performed for each config. - 435 If this is not the case, the resulting idl, as well as the handling - 436 of r_start, r_stop and r_step is wrong and the user has to correct - 437 this in the resulting observable. - 438 - 439 Parameters - 440 ---------- - 441 path : str - 442 Path to .ms.dat files - 443 prefix : str - 444 Ensemble prefix - 445 dtr_read : int - 446 Determines how many trajectories should be skipped - 447 when reading the ms.dat files. - 448 Corresponds to dtr_cnfg / dtr_ms in the openQCD input file. - 449 xmin : int - 450 First timeslice where the boundary - 451 effects have sufficiently decayed. - 452 spatial_extent : int - 453 spatial extent of the lattice, required for normalization. - 454 fit_range : int - 455 Number of data points left and right of the zero - 456 crossing to be included in the linear fit. (Default: 5) - 457 postfix : str - 458 Postfix of measurement file (Default: ms) - 459 c: float - 460 Constant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1. - 461 r_start : list - 462 list which contains the first config to be read for each replicum. - 463 r_stop : list - 464 list which contains the last config to be read for each replicum. - 465 r_step : int - 466 integer that defines a fixed step size between two measurements (in units of configs) - 467 If not given, r_step=1 is assumed. - 468 plaquette : bool - 469 If true extract the plaquette estimate of t0 instead. - 470 names : list - 471 list of names that is assigned to the data according according - 472 to the order in the file list. Use careful, if you do not provide file names! - 473 files : list - 474 list which contains the filenames to be read. No automatic detection of - 475 files performed if given. - 476 plot_fit : bool - 477 If true, the fit for the extraction of t0 is shown together with the data. - 478 assume_thermalization : bool - 479 If True: If the first record divided by the distance between two measurements is larger than - 480 1, it is assumed that this is due to thermalization and the first measurement belongs - 481 to the first config (default). - 482 If False: The config numbers are assumed to be traj_number // difference - 483 - 484 Returns - 485 ------- - 486 t0 : Obs - 487 Extracted t0 - 488 """ - 489 - 490 E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs) - 491 t2E_dict = {} - 492 for t in sorted(E_dict.keys()): - 493 t2E_dict[t] = t ** 2 * E_dict[t] - c - 494 - 495 return fit_t0(t2E_dict, fit_range, plot_fit=kwargs.get('plot_fit')) - 496 - 497 - 498def extract_w0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs): - 499 """Extract w0/a from given .ms.dat files. Returns w0 as Obs. + 428def extract_t0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs): + 429 """Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs. + 430 + 431 It is assumed that all boundary effects have + 432 sufficiently decayed at x0=xmin. + 433 The data around the zero crossing of t^2<E> - c (where c=0.3 by default) + 434 is fitted with a linear function + 435 from which the exact root is extracted. + 436 + 437 It is assumed that one measurement is performed for each config. + 438 If this is not the case, the resulting idl, as well as the handling + 439 of `r_start`, `r_stop` and `r_step` is wrong and the user has to correct + 440 this in the resulting observable. + 441 The function also assumes that `r_step` is the same across all replica. + 442 + 443 Parameters + 444 ---------- + 445 path : str + 446 Path to .ms.dat files + 447 prefix : str + 448 Ensemble prefix + 449 dtr_read : int + 450 Determines how many trajectories should be skipped + 451 when reading the ms.dat files. + 452 Corresponds to dtr_cnfg / dtr_ms in the openQCD input file. + 453 xmin : int + 454 First timeslice where the boundary + 455 effects have sufficiently decayed. + 456 spatial_extent : int + 457 spatial extent of the lattice, required for normalization. + 458 fit_range : int + 459 Number of data points left and right of the zero + 460 crossing to be included in the linear fit. (Default: 5) + 461 postfix : str + 462 Postfix of measurement file (Default: ms) + 463 c: float + 464 Constant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1. + 465 r_start : list + 466 list which contains the first config to be read for each replicum. + 467 r_stop : list + 468 list which contains the last config to be read for each replicum. + 469 r_step : int + 470 integer that defines a fixed step size between two measurements (in units of configs) + 471 If not given, r_step=1 is assumed. + 472 plaquette : bool + 473 If true extract the plaquette estimate of t0 instead. + 474 names : list + 475 list of names that is assigned to the data according according + 476 to the order in the file list. Use careful, if you do not provide file names! + 477 files : list + 478 list which contains the filenames to be read. No automatic detection of + 479 files performed if given. + 480 plot_fit : bool + 481 If true, the fit for the extraction of t0 is shown together with the data. + 482 assume_thermalization : bool + 483 If True: If the first record divided by the distance between two measurements is larger than + 484 1, it is assumed that this is due to thermalization and the first measurement belongs + 485 to the first config (default). + 486 If False: The config numbers are assumed to be traj_number // difference + 487 + 488 Returns + 489 ------- + 490 t0 : Obs + 491 Extracted t0 + 492 """ + 493 + 494 E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs) + 495 t2E_dict = {} + 496 for t in sorted(E_dict.keys()): + 497 t2E_dict[t] = t ** 2 * E_dict[t] - c + 498 + 499 return fit_t0(t2E_dict, fit_range, plot_fit=kwargs.get('plot_fit')) 500 - 501 It is assumed that all boundary effects have - 502 sufficiently decayed at x0=xmin. - 503 The data around the zero crossing of t d(t^2<E>)/dt - (where c=0.3 by default) - 504 is fitted with a linear function - 505 from which the exact root is extracted. - 506 - 507 It is assumed that one measurement is performed for each config. - 508 If this is not the case, the resulting idl, as well as the handling - 509 of r_start, r_stop and r_step is wrong and the user has to correct - 510 this in the resulting observable. - 511 - 512 Parameters - 513 ---------- - 514 path : str - 515 Path to .ms.dat files - 516 prefix : str - 517 Ensemble prefix - 518 dtr_read : int - 519 Determines how many trajectories should be skipped - 520 when reading the ms.dat files. - 521 Corresponds to dtr_cnfg / dtr_ms in the openQCD input file. - 522 xmin : int - 523 First timeslice where the boundary - 524 effects have sufficiently decayed. - 525 spatial_extent : int - 526 spatial extent of the lattice, required for normalization. - 527 fit_range : int - 528 Number of data points left and right of the zero - 529 crossing to be included in the linear fit. (Default: 5) - 530 postfix : str - 531 Postfix of measurement file (Default: ms) - 532 c: float - 533 Constant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1. - 534 r_start : list - 535 list which contains the first config to be read for each replicum. - 536 r_stop : list - 537 list which contains the last config to be read for each replicum. - 538 r_step : int - 539 integer that defines a fixed step size between two measurements (in units of configs) - 540 If not given, r_step=1 is assumed. - 541 plaquette : bool - 542 If true extract the plaquette estimate of w0 instead. - 543 names : list - 544 list of names that is assigned to the data according according - 545 to the order in the file list. Use careful, if you do not provide file names! - 546 files : list - 547 list which contains the filenames to be read. No automatic detection of - 548 files performed if given. - 549 plot_fit : bool - 550 If true, the fit for the extraction of w0 is shown together with the data. - 551 assume_thermalization : bool - 552 If True: If the first record divided by the distance between two measurements is larger than - 553 1, it is assumed that this is due to thermalization and the first measurement belongs - 554 to the first config (default). - 555 If False: The config numbers are assumed to be traj_number // difference - 556 - 557 Returns - 558 ------- - 559 w0 : Obs - 560 Extracted w0 - 561 """ - 562 - 563 E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs) - 564 - 565 ftimes = sorted(E_dict.keys()) + 501 + 502def extract_w0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs): + 503 """Extract w0/a from given .ms.dat files. Returns w0 as Obs. + 504 + 505 It is assumed that all boundary effects have + 506 sufficiently decayed at x0=xmin. + 507 The data around the zero crossing of t d(t^2<E>)/dt - (where c=0.3 by default) + 508 is fitted with a linear function + 509 from which the exact root is extracted. + 510 + 511 It is assumed that one measurement is performed for each config. + 512 If this is not the case, the resulting idl, as well as the handling + 513 of r_start, r_stop and r_step is wrong and the user has to correct + 514 this in the resulting observable. + 515 + 516 Parameters + 517 ---------- + 518 path : str + 519 Path to .ms.dat files + 520 prefix : str + 521 Ensemble prefix + 522 dtr_read : int + 523 Determines how many trajectories should be skipped + 524 when reading the ms.dat files. + 525 Corresponds to dtr_cnfg / dtr_ms in the openQCD input file. + 526 xmin : int + 527 First timeslice where the boundary + 528 effects have sufficiently decayed. + 529 spatial_extent : int + 530 spatial extent of the lattice, required for normalization. + 531 fit_range : int + 532 Number of data points left and right of the zero + 533 crossing to be included in the linear fit. (Default: 5) + 534 postfix : str + 535 Postfix of measurement file (Default: ms) + 536 c: float + 537 Constant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1. + 538 r_start : list + 539 list which contains the first config to be read for each replicum. + 540 r_stop : list + 541 list which contains the last config to be read for each replicum. + 542 r_step : int + 543 integer that defines a fixed step size between two measurements (in units of configs) + 544 If not given, r_step=1 is assumed. + 545 plaquette : bool + 546 If true extract the plaquette estimate of w0 instead. + 547 names : list + 548 list of names that is assigned to the data according according + 549 to the order in the file list. Use careful, if you do not provide file names! + 550 files : list + 551 list which contains the filenames to be read. No automatic detection of + 552 files performed if given. + 553 plot_fit : bool + 554 If true, the fit for the extraction of w0 is shown together with the data. + 555 assume_thermalization : bool + 556 If True: If the first record divided by the distance between two measurements is larger than + 557 1, it is assumed that this is due to thermalization and the first measurement belongs + 558 to the first config (default). + 559 If False: The config numbers are assumed to be traj_number // difference + 560 + 561 Returns + 562 ------- + 563 w0 : Obs + 564 Extracted w0 + 565 """ 566 - 567 t2E_dict = {} - 568 for t in ftimes: - 569 t2E_dict[t] = t ** 2 * E_dict[t] + 567 E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs) + 568 + 569 ftimes = sorted(E_dict.keys()) 570 - 571 tdtt2E_dict = {} - 572 tdtt2E_dict[ftimes[0]] = ftimes[0] * (t2E_dict[ftimes[1]] - t2E_dict[ftimes[0]]) / (ftimes[1] - ftimes[0]) - c - 573 for i in range(1, len(ftimes) - 1): - 574 tdtt2E_dict[ftimes[i]] = ftimes[i] * (t2E_dict[ftimes[i + 1]] - t2E_dict[ftimes[i - 1]]) / (ftimes[i + 1] - ftimes[i - 1]) - c - 575 tdtt2E_dict[ftimes[-1]] = ftimes[-1] * (t2E_dict[ftimes[-1]] - t2E_dict[ftimes[-2]]) / (ftimes[-1] - ftimes[-2]) - c - 576 - 577 return np.sqrt(fit_t0(tdtt2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'), observable='w0')) - 578 - 579 - 580def _parse_array_openQCD2(d, n, size, wa, quadrupel=False): - 581 arr = [] - 582 if d == 2: - 583 for i in range(n[0]): - 584 tmp = wa[i * n[1]:(i + 1) * n[1]] - 585 if quadrupel: - 586 tmp2 = [] - 587 for j in range(0, len(tmp), 2): - 588 tmp2.append(tmp[j]) - 589 arr.append(tmp2) - 590 else: - 591 arr.append(np.asarray(tmp)) - 592 - 593 else: - 594 raise Exception('Only two-dimensional arrays supported!') - 595 - 596 return arr - 597 - 598 - 599def _find_files(path, prefix, postfix, ext, known_files=[]): - 600 found = [] - 601 files = [] + 571 t2E_dict = {} + 572 for t in ftimes: + 573 t2E_dict[t] = t ** 2 * E_dict[t] + 574 + 575 tdtt2E_dict = {} + 576 tdtt2E_dict[ftimes[0]] = ftimes[0] * (t2E_dict[ftimes[1]] - t2E_dict[ftimes[0]]) / (ftimes[1] - ftimes[0]) - c + 577 for i in range(1, len(ftimes) - 1): + 578 tdtt2E_dict[ftimes[i]] = ftimes[i] * (t2E_dict[ftimes[i + 1]] - t2E_dict[ftimes[i - 1]]) / (ftimes[i + 1] - ftimes[i - 1]) - c + 579 tdtt2E_dict[ftimes[-1]] = ftimes[-1] * (t2E_dict[ftimes[-1]] - t2E_dict[ftimes[-2]]) / (ftimes[-1] - ftimes[-2]) - c + 580 + 581 return np.sqrt(fit_t0(tdtt2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'), observable='w0')) + 582 + 583 + 584def _parse_array_openQCD2(d, n, size, wa, quadrupel=False): + 585 arr = [] + 586 if d == 2: + 587 for i in range(n[0]): + 588 tmp = wa[i * n[1]:(i + 1) * n[1]] + 589 if quadrupel: + 590 tmp2 = [] + 591 for j in range(0, len(tmp), 2): + 592 tmp2.append(tmp[j]) + 593 arr.append(tmp2) + 594 else: + 595 arr.append(np.asarray(tmp)) + 596 + 597 else: + 598 raise Exception('Only two-dimensional arrays supported!') + 599 + 600 return arr + 601 602 - 603 if postfix != "": - 604 if postfix[-1] != ".": - 605 postfix = postfix + "." - 606 if postfix[0] != ".": - 607 postfix = "." + postfix - 608 - 609 if ext[0] == ".": - 610 ext = ext[1:] - 611 - 612 pattern = prefix + "*" + postfix + ext - 613 - 614 for (dirpath, dirnames, filenames) in os.walk(path + "/"): - 615 found.extend(filenames) - 616 break + 603def _find_files(path, prefix, postfix, ext, known_files=[]): + 604 found = [] + 605 files = [] + 606 + 607 if postfix != "": + 608 if postfix[-1] != ".": + 609 postfix = postfix + "." + 610 if postfix[0] != ".": + 611 postfix = "." + postfix + 612 + 613 if ext[0] == ".": + 614 ext = ext[1:] + 615 + 616 pattern = prefix + "*" + postfix + ext 617 - 618 if known_files != []: - 619 for kf in known_files: - 620 if kf not in found: - 621 raise FileNotFoundError("Given file " + kf + " does not exist!") - 622 - 623 return known_files - 624 - 625 if not found: - 626 raise FileNotFoundError(f"Error, directory '{path}' not found") - 627 - 628 for f in found: - 629 if fnmatch.fnmatch(f, pattern): - 630 files.append(f) + 618 for (dirpath, dirnames, filenames) in os.walk(path + "/"): + 619 found.extend(filenames) + 620 break + 621 + 622 if known_files != []: + 623 for kf in known_files: + 624 if kf not in found: + 625 raise FileNotFoundError("Given file " + kf + " does not exist!") + 626 + 627 return known_files + 628 + 629 if not found: + 630 raise FileNotFoundError(f"Error, directory '{path}' not found") 631 - 632 if files == []: - 633 raise Exception("No files found after pattern filter!") - 634 - 635 files = sort_names(files) - 636 return files - 637 + 632 for f in found: + 633 if fnmatch.fnmatch(f, pattern): + 634 files.append(f) + 635 + 636 if files == []: + 637 raise Exception("No files found after pattern filter!") 638 - 639def _read_array_openQCD2(fp): - 640 t = fp.read(4) - 641 d = struct.unpack('i', t)[0] - 642 t = fp.read(4 * d) - 643 n = struct.unpack('%di' % (d), t) + 639 files = sort_names(files) + 640 return files + 641 + 642 + 643def _read_array_openQCD2(fp): 644 t = fp.read(4) - 645 size = struct.unpack('i', t)[0] - 646 if size == 4: - 647 types = 'i' - 648 elif size == 8: - 649 types = 'd' - 650 elif size == 16: - 651 types = 'dd' - 652 else: - 653 raise Exception("Type for size '" + str(size) + "' not known.") - 654 m = n[0] - 655 for i in range(1, d): - 656 m *= n[i] - 657 - 658 t = fp.read(m * size) - 659 tmp = struct.unpack('%d%s' % (m, types), t) - 660 - 661 arr = _parse_array_openQCD2(d, n, size, tmp, quadrupel=True) - 662 return {'d': d, 'n': n, 'size': size, 'arr': arr} - 663 + 645 d = struct.unpack('i', t)[0] + 646 t = fp.read(4 * d) + 647 n = struct.unpack('%di' % (d), t) + 648 t = fp.read(4) + 649 size = struct.unpack('i', t)[0] + 650 if size == 4: + 651 types = 'i' + 652 elif size == 8: + 653 types = 'd' + 654 elif size == 16: + 655 types = 'dd' + 656 else: + 657 raise Exception("Type for size '" + str(size) + "' not known.") + 658 m = n[0] + 659 for i in range(1, d): + 660 m *= n[i] + 661 + 662 t = fp.read(m * size) + 663 tmp = struct.unpack('%d%s' % (m, types), t) 664 - 665def read_qtop(path, prefix, c, dtr_cnfg=1, version="openQCD", **kwargs): - 666 """Read the topologial charge based on openQCD gradient flow measurements. + 665 arr = _parse_array_openQCD2(d, n, size, tmp, quadrupel=True) + 666 return {'d': d, 'n': n, 'size': size, 'arr': arr} 667 - 668 Parameters - 669 ---------- - 670 path : str - 671 path of the measurement files - 672 prefix : str - 673 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. - 674 Ignored if file names are passed explicitly via keyword files. - 675 c : double - 676 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. - 677 dtr_cnfg : int - 678 (optional) parameter that specifies the number of measurements - 679 between two configs. - 680 If it is not set, the distance between two measurements - 681 in the file is assumed to be the distance between two configurations. - 682 steps : int - 683 (optional) Distance between two configurations in units of trajectories / - 684 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given - 685 version : str - 686 Either openQCD or sfqcd, depending on the data. - 687 L : int - 688 spatial length of the lattice in L/a. - 689 HAS to be set if version != sfqcd, since openQCD does not provide - 690 this in the header - 691 r_start : list - 692 list which contains the first config to be read for each replicum. - 693 r_stop : list - 694 list which contains the last config to be read for each replicum. - 695 files : list - 696 specify the exact files that need to be read - 697 from path, practical if e.g. only one replicum is needed - 698 postfix : str - 699 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files - 700 names : list - 701 Alternative labeling for replicas/ensembles. - 702 Has to have the appropriate length. - 703 Zeuthen_flow : bool - 704 (optional) If True, the Zeuthen flow is used for Qtop. Only possible - 705 for version=='sfqcd' If False, the Wilson flow is used. - 706 integer_charge : bool - 707 If True, the charge is rounded towards the nearest integer on each config. - 708 - 709 Returns - 710 ------- - 711 result : Obs - 712 Read topological charge - 713 """ - 714 - 715 return _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version=version, obspos=0, **kwargs) - 716 - 717 - 718def read_gf_coupling(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs): - 719 """Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details. + 668 + 669def read_qtop(path, prefix, c, dtr_cnfg=1, version="openQCD", **kwargs): + 670 """Read the topologial charge based on openQCD gradient flow measurements. + 671 + 672 Parameters + 673 ---------- + 674 path : str + 675 path of the measurement files + 676 prefix : str + 677 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. + 678 Ignored if file names are passed explicitly via keyword files. + 679 c : double + 680 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. + 681 dtr_cnfg : int + 682 (optional) parameter that specifies the number of measurements + 683 between two configs. + 684 If it is not set, the distance between two measurements + 685 in the file is assumed to be the distance between two configurations. + 686 steps : int + 687 (optional) Distance between two configurations in units of trajectories / + 688 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given + 689 version : str + 690 Either openQCD or sfqcd, depending on the data. + 691 L : int + 692 spatial length of the lattice in L/a. + 693 HAS to be set if version != sfqcd, since openQCD does not provide + 694 this in the header + 695 r_start : list + 696 list which contains the first config to be read for each replicum. + 697 r_stop : list + 698 list which contains the last config to be read for each replicum. + 699 files : list + 700 specify the exact files that need to be read + 701 from path, practical if e.g. only one replicum is needed + 702 postfix : str + 703 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files + 704 names : list + 705 Alternative labeling for replicas/ensembles. + 706 Has to have the appropriate length. + 707 Zeuthen_flow : bool + 708 (optional) If True, the Zeuthen flow is used for Qtop. Only possible + 709 for version=='sfqcd' If False, the Wilson flow is used. + 710 integer_charge : bool + 711 If True, the charge is rounded towards the nearest integer on each config. + 712 + 713 Returns + 714 ------- + 715 result : Obs + 716 Read topological charge + 717 """ + 718 + 719 return _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version=version, obspos=0, **kwargs) 720 - 721 Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step. - 722 - 723 Parameters - 724 ---------- - 725 path : str - 726 path of the measurement files - 727 prefix : str - 728 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. - 729 Ignored if file names are passed explicitly via keyword files. - 730 c : double - 731 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. - 732 dtr_cnfg : int - 733 (optional) parameter that specifies the number of measurements - 734 between two configs. - 735 If it is not set, the distance between two measurements - 736 in the file is assumed to be the distance between two configurations. - 737 steps : int - 738 (optional) Distance between two configurations in units of trajectories / - 739 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given - 740 r_start : list - 741 list which contains the first config to be read for each replicum. - 742 r_stop : list - 743 list which contains the last config to be read for each replicum. - 744 files : list - 745 specify the exact files that need to be read - 746 from path, practical if e.g. only one replicum is needed - 747 names : list - 748 Alternative labeling for replicas/ensembles. - 749 Has to have the appropriate length. - 750 postfix : str - 751 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files - 752 Zeuthen_flow : bool - 753 (optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used. - 754 """ - 755 - 756 if c != 0.3: - 757 raise Exception("The required lattice norm is only implemented for c=0.3 at the moment.") - 758 - 759 plaq = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=6, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs) - 760 C2x1 = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=7, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs) - 761 L = plaq.tag["L"] - 762 T = plaq.tag["T"] - 763 - 764 if T != L: - 765 raise Exception("The required lattice norm is only implemented for T=L at the moment.") - 766 - 767 if Zeuthen_flow is not True: - 768 raise Exception("The required lattice norm is only implemented for the Zeuthen flow at the moment.") - 769 - 770 t = (c * L) ** 2 / 8 - 771 - 772 normdict = {4: 0.012341170468270, - 773 6: 0.010162691462430, - 774 8: 0.009031614807931, - 775 10: 0.008744966371393, - 776 12: 0.008650917856809, - 777 14: 8.611154391267955E-03, - 778 16: 0.008591758449508, - 779 20: 0.008575359627103, - 780 24: 0.008569387847540, - 781 28: 8.566803713382559E-03, - 782 32: 0.008565541650006, - 783 40: 8.564480684962046E-03, - 784 48: 8.564098025073460E-03, - 785 64: 8.563853943383087E-03} - 786 - 787 return t * t * (5 / 3 * plaq - 1 / 12 * C2x1) / normdict[L] - 788 - 789 - 790def _read_flow_obs(path, prefix, c, dtr_cnfg=1, version="openQCD", obspos=0, sum_t=True, **kwargs): - 791 """Read a flow observable based on openQCD gradient flow measurements. + 721 + 722def read_gf_coupling(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs): + 723 """Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details. + 724 + 725 Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step. + 726 + 727 Parameters + 728 ---------- + 729 path : str + 730 path of the measurement files + 731 prefix : str + 732 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. + 733 Ignored if file names are passed explicitly via keyword files. + 734 c : double + 735 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. + 736 dtr_cnfg : int + 737 (optional) parameter that specifies the number of measurements + 738 between two configs. + 739 If it is not set, the distance between two measurements + 740 in the file is assumed to be the distance between two configurations. + 741 steps : int + 742 (optional) Distance between two configurations in units of trajectories / + 743 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given + 744 r_start : list + 745 list which contains the first config to be read for each replicum. + 746 r_stop : list + 747 list which contains the last config to be read for each replicum. + 748 files : list + 749 specify the exact files that need to be read + 750 from path, practical if e.g. only one replicum is needed + 751 names : list + 752 Alternative labeling for replicas/ensembles. + 753 Has to have the appropriate length. + 754 postfix : str + 755 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files + 756 Zeuthen_flow : bool + 757 (optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used. + 758 """ + 759 + 760 if c != 0.3: + 761 raise Exception("The required lattice norm is only implemented for c=0.3 at the moment.") + 762 + 763 plaq = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=6, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs) + 764 C2x1 = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=7, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs) + 765 L = plaq.tag["L"] + 766 T = plaq.tag["T"] + 767 + 768 if T != L: + 769 raise Exception("The required lattice norm is only implemented for T=L at the moment.") + 770 + 771 if Zeuthen_flow is not True: + 772 raise Exception("The required lattice norm is only implemented for the Zeuthen flow at the moment.") + 773 + 774 t = (c * L) ** 2 / 8 + 775 + 776 normdict = {4: 0.012341170468270, + 777 6: 0.010162691462430, + 778 8: 0.009031614807931, + 779 10: 0.008744966371393, + 780 12: 0.008650917856809, + 781 14: 8.611154391267955E-03, + 782 16: 0.008591758449508, + 783 20: 0.008575359627103, + 784 24: 0.008569387847540, + 785 28: 8.566803713382559E-03, + 786 32: 0.008565541650006, + 787 40: 8.564480684962046E-03, + 788 48: 8.564098025073460E-03, + 789 64: 8.563853943383087E-03} + 790 + 791 return t * t * (5 / 3 * plaq - 1 / 12 * C2x1) / normdict[L] 792 - 793 Parameters - 794 ---------- - 795 path : str - 796 path of the measurement files - 797 prefix : str - 798 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. - 799 Ignored if file names are passed explicitly via keyword files. - 800 c : double - 801 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. - 802 dtr_cnfg : int - 803 (optional) parameter that specifies the number of measurements - 804 between two configs. - 805 If it is not set, the distance between two measurements - 806 in the file is assumed to be the distance between two configurations. - 807 steps : int - 808 (optional) Distance between two configurations in units of trajectories / - 809 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given - 810 version : str - 811 Either openQCD or sfqcd, depending on the data. - 812 obspos : int - 813 position of the obeservable in the measurement file. Only relevant for sfqcd files. - 814 sum_t : bool - 815 If true sum over all timeslices, if false only take the value at T/2. - 816 L : int - 817 spatial length of the lattice in L/a. - 818 HAS to be set if version != sfqcd, since openQCD does not provide - 819 this in the header - 820 r_start : list - 821 list which contains the first config to be read for each replicum. - 822 r_stop : list - 823 list which contains the last config to be read for each replicum. - 824 files : list - 825 specify the exact files that need to be read - 826 from path, practical if e.g. only one replicum is needed - 827 names : list - 828 Alternative labeling for replicas/ensembles. - 829 Has to have the appropriate length. - 830 postfix : str - 831 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files - 832 Zeuthen_flow : bool - 833 (optional) If True, the Zeuthen flow is used for Qtop. Only possible - 834 for version=='sfqcd' If False, the Wilson flow is used. - 835 integer_charge : bool - 836 If True, the charge is rounded towards the nearest integer on each config. - 837 - 838 Returns - 839 ------- - 840 result : Obs - 841 flow observable specified - 842 """ - 843 known_versions = ["openQCD", "sfqcd"] - 844 - 845 if version not in known_versions: - 846 raise Exception("Unknown openQCD version.") - 847 if "steps" in kwargs: - 848 steps = kwargs.get("steps") - 849 if version == "sfqcd": - 850 if "L" in kwargs: - 851 supposed_L = kwargs.get("L") - 852 else: - 853 supposed_L = None - 854 postfix = "gfms" - 855 else: - 856 if "L" not in kwargs: - 857 raise Exception("This version of openQCD needs you to provide the spatial length of the lattice as parameter 'L'.") - 858 else: - 859 L = kwargs.get("L") - 860 postfix = "ms" - 861 - 862 if "postfix" in kwargs: - 863 postfix = kwargs.get("postfix") - 864 - 865 if "files" in kwargs: - 866 known_files = kwargs.get("files") - 867 else: - 868 known_files = [] - 869 - 870 files = _find_files(path, prefix, postfix, "dat", known_files=known_files) - 871 - 872 if 'r_start' in kwargs: - 873 r_start = kwargs.get('r_start') - 874 if len(r_start) != len(files): - 875 raise Exception('r_start does not match number of replicas') - 876 r_start = [o if o else None for o in r_start] - 877 else: - 878 r_start = [None] * len(files) - 879 - 880 if 'r_stop' in kwargs: - 881 r_stop = kwargs.get('r_stop') - 882 if len(r_stop) != len(files): - 883 raise Exception('r_stop does not match number of replicas') - 884 else: - 885 r_stop = [None] * len(files) - 886 rep_names = [] - 887 - 888 zeuthen = kwargs.get('Zeuthen_flow', False) - 889 if zeuthen and version not in ['sfqcd']: - 890 raise Exception('Zeuthen flow can only be used for version==sfqcd') + 793 + 794def _read_flow_obs(path, prefix, c, dtr_cnfg=1, version="openQCD", obspos=0, sum_t=True, **kwargs): + 795 """Read a flow observable based on openQCD gradient flow measurements. + 796 + 797 Parameters + 798 ---------- + 799 path : str + 800 path of the measurement files + 801 prefix : str + 802 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat. + 803 Ignored if file names are passed explicitly via keyword files. + 804 c : double + 805 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L. + 806 dtr_cnfg : int + 807 (optional) parameter that specifies the number of measurements + 808 between two configs. + 809 If it is not set, the distance between two measurements + 810 in the file is assumed to be the distance between two configurations. + 811 steps : int + 812 (optional) Distance between two configurations in units of trajectories / + 813 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given + 814 version : str + 815 Either openQCD or sfqcd, depending on the data. + 816 obspos : int + 817 position of the obeservable in the measurement file. Only relevant for sfqcd files. + 818 sum_t : bool + 819 If true sum over all timeslices, if false only take the value at T/2. + 820 L : int + 821 spatial length of the lattice in L/a. + 822 HAS to be set if version != sfqcd, since openQCD does not provide + 823 this in the header + 824 r_start : list + 825 list which contains the first config to be read for each replicum. + 826 r_stop : list + 827 list which contains the last config to be read for each replicum. + 828 files : list + 829 specify the exact files that need to be read + 830 from path, practical if e.g. only one replicum is needed + 831 names : list + 832 Alternative labeling for replicas/ensembles. + 833 Has to have the appropriate length. + 834 postfix : str + 835 postfix of the file to read, e.g. '.gfms.dat' for openQCD-files + 836 Zeuthen_flow : bool + 837 (optional) If True, the Zeuthen flow is used for Qtop. Only possible + 838 for version=='sfqcd' If False, the Wilson flow is used. + 839 integer_charge : bool + 840 If True, the charge is rounded towards the nearest integer on each config. + 841 + 842 Returns + 843 ------- + 844 result : Obs + 845 flow observable specified + 846 """ + 847 known_versions = ["openQCD", "sfqcd"] + 848 + 849 if version not in known_versions: + 850 raise Exception("Unknown openQCD version.") + 851 if "steps" in kwargs: + 852 steps = kwargs.get("steps") + 853 if version == "sfqcd": + 854 if "L" in kwargs: + 855 supposed_L = kwargs.get("L") + 856 else: + 857 supposed_L = None + 858 postfix = "gfms" + 859 else: + 860 if "L" not in kwargs: + 861 raise Exception("This version of openQCD needs you to provide the spatial length of the lattice as parameter 'L'.") + 862 else: + 863 L = kwargs.get("L") + 864 postfix = "ms" + 865 + 866 if "postfix" in kwargs: + 867 postfix = kwargs.get("postfix") + 868 + 869 if "files" in kwargs: + 870 known_files = kwargs.get("files") + 871 else: + 872 known_files = [] + 873 + 874 files = _find_files(path, prefix, postfix, "dat", known_files=known_files) + 875 + 876 if 'r_start' in kwargs: + 877 r_start = kwargs.get('r_start') + 878 if len(r_start) != len(files): + 879 raise Exception('r_start does not match number of replicas') + 880 r_start = [o if o else None for o in r_start] + 881 else: + 882 r_start = [None] * len(files) + 883 + 884 if 'r_stop' in kwargs: + 885 r_stop = kwargs.get('r_stop') + 886 if len(r_stop) != len(files): + 887 raise Exception('r_stop does not match number of replicas') + 888 else: + 889 r_stop = [None] * len(files) + 890 rep_names = [] 891 - 892 r_start_index = [] - 893 r_stop_index = [] - 894 deltas = [] - 895 configlist = [] - 896 if not zeuthen: - 897 obspos += 8 - 898 for rep, file in enumerate(files): - 899 with open(path + "/" + file, "rb") as fp: - 900 - 901 Q = [] - 902 traj_list = [] - 903 if version in ['sfqcd']: - 904 t = fp.read(12) - 905 header = struct.unpack('<iii', t) - 906 zthfl = header[0] # Zeuthen flow -> if it's equal to 2 it means that the Zeuthen flow is also 'measured' (apart from the Wilson flow) - 907 ncs = header[1] # number of different values for c in t_flow=1/8 c² L² -> measurements done for ncs c's - 908 tmax = header[2] # lattice T/a - 909 - 910 t = fp.read(12) - 911 Ls = struct.unpack('<iii', t) - 912 if (Ls[0] == Ls[1] and Ls[1] == Ls[2]): - 913 L = Ls[0] - 914 if not (supposed_L == L) and supposed_L: - 915 raise Exception("It seems the length given in the header and by you contradict each other") - 916 else: - 917 raise Exception("Found more than one spatial length in header!") - 918 - 919 t = fp.read(16) - 920 header2 = struct.unpack('<dd', t) - 921 tol = header2[0] - 922 cmax = header2[1] # highest value of c used - 923 - 924 if c > cmax: - 925 raise Exception('Flow has been determined between c=0 and c=%lf with tolerance %lf' % (cmax, tol)) - 926 - 927 if (zthfl == 2): - 928 nfl = 2 # number of flows - 929 else: - 930 nfl = 1 - 931 iobs = 8 * nfl # number of flow observables calculated - 932 - 933 while True: - 934 t = fp.read(4) - 935 if (len(t) < 4): - 936 break - 937 traj_list.append(struct.unpack('i', t)[0]) # trajectory number when measurement was done - 938 - 939 for j in range(ncs + 1): - 940 for i in range(iobs): - 941 t = fp.read(8 * tmax) - 942 if (i == obspos): # determines the flow observable -> i=0 <-> Zeuthen flow - 943 Q.append(struct.unpack('d' * tmax, t)) - 944 - 945 else: - 946 t = fp.read(12) - 947 header = struct.unpack('<iii', t) - 948 # step size in integration steps "dnms" - 949 dn = header[0] - 950 # number of measurements, so "ntot"/dn - 951 nn = header[1] - 952 # lattice T/a - 953 tmax = header[2] - 954 - 955 t = fp.read(8) - 956 eps = struct.unpack('d', t)[0] - 957 - 958 while True: - 959 t = fp.read(4) - 960 if (len(t) < 4): - 961 break - 962 traj_list.append(struct.unpack('i', t)[0]) - 963 # Wsl - 964 t = fp.read(8 * tmax * (nn + 1)) - 965 # Ysl - 966 t = fp.read(8 * tmax * (nn + 1)) - 967 # Qsl, which is asked for in this method + 892 zeuthen = kwargs.get('Zeuthen_flow', False) + 893 if zeuthen and version not in ['sfqcd']: + 894 raise Exception('Zeuthen flow can only be used for version==sfqcd') + 895 + 896 r_start_index = [] + 897 r_stop_index = [] + 898 deltas = [] + 899 configlist = [] + 900 if not zeuthen: + 901 obspos += 8 + 902 for rep, file in enumerate(files): + 903 with open(path + "/" + file, "rb") as fp: + 904 + 905 Q = [] + 906 traj_list = [] + 907 if version in ['sfqcd']: + 908 t = fp.read(12) + 909 header = struct.unpack('<iii', t) + 910 zthfl = header[0] # Zeuthen flow -> if it's equal to 2 it means that the Zeuthen flow is also 'measured' (apart from the Wilson flow) + 911 ncs = header[1] # number of different values for c in t_flow=1/8 c² L² -> measurements done for ncs c's + 912 tmax = header[2] # lattice T/a + 913 + 914 t = fp.read(12) + 915 Ls = struct.unpack('<iii', t) + 916 if (Ls[0] == Ls[1] and Ls[1] == Ls[2]): + 917 L = Ls[0] + 918 if not (supposed_L == L) and supposed_L: + 919 raise Exception("It seems the length given in the header and by you contradict each other") + 920 else: + 921 raise Exception("Found more than one spatial length in header!") + 922 + 923 t = fp.read(16) + 924 header2 = struct.unpack('<dd', t) + 925 tol = header2[0] + 926 cmax = header2[1] # highest value of c used + 927 + 928 if c > cmax: + 929 raise Exception('Flow has been determined between c=0 and c=%lf with tolerance %lf' % (cmax, tol)) + 930 + 931 if (zthfl == 2): + 932 nfl = 2 # number of flows + 933 else: + 934 nfl = 1 + 935 iobs = 8 * nfl # number of flow observables calculated + 936 + 937 while True: + 938 t = fp.read(4) + 939 if (len(t) < 4): + 940 break + 941 traj_list.append(struct.unpack('i', t)[0]) # trajectory number when measurement was done + 942 + 943 for j in range(ncs + 1): + 944 for i in range(iobs): + 945 t = fp.read(8 * tmax) + 946 if (i == obspos): # determines the flow observable -> i=0 <-> Zeuthen flow + 947 Q.append(struct.unpack('d' * tmax, t)) + 948 + 949 else: + 950 t = fp.read(12) + 951 header = struct.unpack('<iii', t) + 952 # step size in integration steps "dnms" + 953 dn = header[0] + 954 # number of measurements, so "ntot"/dn + 955 nn = header[1] + 956 # lattice T/a + 957 tmax = header[2] + 958 + 959 t = fp.read(8) + 960 eps = struct.unpack('d', t)[0] + 961 + 962 while True: + 963 t = fp.read(4) + 964 if (len(t) < 4): + 965 break + 966 traj_list.append(struct.unpack('i', t)[0]) + 967 # Wsl 968 t = fp.read(8 * tmax * (nn + 1)) - 969 # unpack the array of Qtops, - 970 # on each timeslice t=0,...,tmax-1 and the - 971 # measurement number in = 0...nn (see README.qcd1) - 972 tmpd = struct.unpack('d' * tmax * (nn + 1), t) - 973 Q.append(tmpd) - 974 - 975 if len(np.unique(np.diff(traj_list))) != 1: - 976 raise Exception("Irregularities in stepsize found") - 977 else: - 978 if 'steps' in kwargs: - 979 if steps != traj_list[1] - traj_list[0]: - 980 raise Exception("steps and the found stepsize are not the same") - 981 else: - 982 steps = traj_list[1] - traj_list[0] - 983 - 984 configlist.append([tr // steps // dtr_cnfg for tr in traj_list]) - 985 if configlist[-1][0] > 1: - 986 offset = configlist[-1][0] - 1 - 987 warnings.warn('Assume thermalization and that the first measurement belongs to the first config. Offset = %d configs (%d trajectories / cycles)' % ( - 988 offset, offset * steps)) - 989 configlist[-1] = [item - offset for item in configlist[-1]] - 990 - 991 if r_start[rep] is None: - 992 r_start_index.append(0) - 993 else: - 994 try: - 995 r_start_index.append(configlist[-1].index(r_start[rep])) - 996 except ValueError: - 997 raise Exception('Config %d not in file with range [%d, %d]' % ( - 998 r_start[rep], configlist[-1][0], configlist[-1][-1])) from None - 999 -1000 if r_stop[rep] is None: -1001 r_stop_index.append(len(configlist[-1]) - 1) -1002 else: -1003 try: -1004 r_stop_index.append(configlist[-1].index(r_stop[rep])) -1005 except ValueError: -1006 raise Exception('Config %d not in file with range [%d, %d]' % ( -1007 r_stop[rep], configlist[-1][0], configlist[-1][-1])) from None -1008 -1009 if version in ['sfqcd']: -1010 cstepsize = cmax / ncs -1011 index_aim = round(c / cstepsize) -1012 else: -1013 t_aim = (c * L) ** 2 / 8 -1014 index_aim = round(t_aim / eps / dn) -1015 -1016 Q_sum = [] -1017 for i, item in enumerate(Q): -1018 if sum_t is True: -1019 Q_sum.append([sum(item[current:current + tmax]) -1020 for current in range(0, len(item), tmax)]) -1021 else: -1022 Q_sum.append([item[int(tmax / 2)]]) -1023 Q_top = [] -1024 if version in ['sfqcd']: -1025 for i in range(len(Q_sum) // (ncs + 1)): -1026 Q_top.append(Q_sum[i * (ncs + 1) + index_aim][0]) -1027 else: -1028 for i in range(len(Q) // dtr_cnfg): -1029 Q_top.append(Q_sum[dtr_cnfg * i][index_aim]) -1030 if len(Q_top) != len(traj_list) // dtr_cnfg: -1031 raise Exception("qtops and traj_list dont have the same length") -1032 -1033 if kwargs.get('integer_charge', False): -1034 Q_top = [round(q) for q in Q_top] -1035 -1036 truncated_file = file[:-len(postfix)] -1037 -1038 if "names" not in kwargs: -1039 try: -1040 idx = truncated_file.index('r') -1041 except Exception: -1042 if "names" not in kwargs: -1043 raise Exception("Automatic recognition of replicum failed, please enter the key word 'names'.") -1044 ens_name = truncated_file[:idx] -1045 rep_names.append(ens_name + '|' + truncated_file[idx:].split(".")[0]) -1046 else: -1047 names = kwargs.get("names") -1048 rep_names = names -1049 -1050 deltas.append(Q_top) -1051 -1052 rep_names = sort_names(rep_names) + 969 # Ysl + 970 t = fp.read(8 * tmax * (nn + 1)) + 971 # Qsl, which is asked for in this method + 972 t = fp.read(8 * tmax * (nn + 1)) + 973 # unpack the array of Qtops, + 974 # on each timeslice t=0,...,tmax-1 and the + 975 # measurement number in = 0...nn (see README.qcd1) + 976 tmpd = struct.unpack('d' * tmax * (nn + 1), t) + 977 Q.append(tmpd) + 978 + 979 if len(np.unique(np.diff(traj_list))) != 1: + 980 raise Exception("Irregularities in stepsize found") + 981 else: + 982 if 'steps' in kwargs: + 983 if steps != traj_list[1] - traj_list[0]: + 984 raise Exception("steps and the found stepsize are not the same") + 985 else: + 986 steps = traj_list[1] - traj_list[0] + 987 + 988 configlist.append([tr // steps // dtr_cnfg for tr in traj_list]) + 989 if configlist[-1][0] > 1: + 990 offset = configlist[-1][0] - 1 + 991 warnings.warn('Assume thermalization and that the first measurement belongs to the first config. Offset = %d configs (%d trajectories / cycles)' % ( + 992 offset, offset * steps)) + 993 configlist[-1] = [item - offset for item in configlist[-1]] + 994 + 995 if r_start[rep] is None: + 996 r_start_index.append(0) + 997 else: + 998 try: + 999 r_start_index.append(configlist[-1].index(r_start[rep])) +1000 except ValueError: +1001 raise Exception('Config %d not in file with range [%d, %d]' % ( +1002 r_start[rep], configlist[-1][0], configlist[-1][-1])) from None +1003 +1004 if r_stop[rep] is None: +1005 r_stop_index.append(len(configlist[-1]) - 1) +1006 else: +1007 try: +1008 r_stop_index.append(configlist[-1].index(r_stop[rep])) +1009 except ValueError: +1010 raise Exception('Config %d not in file with range [%d, %d]' % ( +1011 r_stop[rep], configlist[-1][0], configlist[-1][-1])) from None +1012 +1013 if version in ['sfqcd']: +1014 cstepsize = cmax / ncs +1015 index_aim = round(c / cstepsize) +1016 else: +1017 t_aim = (c * L) ** 2 / 8 +1018 index_aim = round(t_aim / eps / dn) +1019 +1020 Q_sum = [] +1021 for i, item in enumerate(Q): +1022 if sum_t is True: +1023 Q_sum.append([sum(item[current:current + tmax]) +1024 for current in range(0, len(item), tmax)]) +1025 else: +1026 Q_sum.append([item[int(tmax / 2)]]) +1027 Q_top = [] +1028 if version in ['sfqcd']: +1029 for i in range(len(Q_sum) // (ncs + 1)): +1030 Q_top.append(Q_sum[i * (ncs + 1) + index_aim][0]) +1031 else: +1032 for i in range(len(Q) // dtr_cnfg): +1033 Q_top.append(Q_sum[dtr_cnfg * i][index_aim]) +1034 if len(Q_top) != len(traj_list) // dtr_cnfg: +1035 raise Exception("qtops and traj_list dont have the same length") +1036 +1037 if kwargs.get('integer_charge', False): +1038 Q_top = [round(q) for q in Q_top] +1039 +1040 truncated_file = file[:-len(postfix)] +1041 +1042 if "names" not in kwargs: +1043 try: +1044 idx = truncated_file.index('r') +1045 except Exception: +1046 if "names" not in kwargs: +1047 raise Exception("Automatic recognition of replicum failed, please enter the key word 'names'.") +1048 ens_name = truncated_file[:idx] +1049 rep_names.append(ens_name + '|' + truncated_file[idx:].split(".")[0]) +1050 else: +1051 names = kwargs.get("names") +1052 rep_names = names 1053 -1054 idl = [range(int(configlist[rep][r_start_index[rep]]), int(configlist[rep][r_stop_index[rep]]) + 1, 1) for rep in range(len(deltas))] -1055 deltas = [deltas[nrep][r_start_index[nrep]:r_stop_index[nrep] + 1] for nrep in range(len(deltas))] -1056 result = Obs(deltas, rep_names, idl=idl) -1057 result.tag = {"T": tmax - 1, -1058 "L": L} -1059 return result -1060 -1061 -1062def qtop_projection(qtop, target=0): -1063 """Returns the projection to the topological charge sector defined by target. +1054 deltas.append(Q_top) +1055 +1056 rep_names = sort_names(rep_names) +1057 +1058 idl = [range(int(configlist[rep][r_start_index[rep]]), int(configlist[rep][r_stop_index[rep]]) + 1, 1) for rep in range(len(deltas))] +1059 deltas = [deltas[nrep][r_start_index[nrep]:r_stop_index[nrep] + 1] for nrep in range(len(deltas))] +1060 result = Obs(deltas, rep_names, idl=idl) +1061 result.tag = {"T": tmax - 1, +1062 "L": L} +1063 return result 1064 -1065 Parameters -1066 ---------- -1067 path : Obs -1068 Topological charge. -1069 target : int -1070 Specifies the topological sector to be reweighted to (default 0) -1071 -1072 Returns -1073 ------- -1074 reto : Obs -1075 projection to the topological charge sector defined by target -1076 """ -1077 if qtop.reweighted: -1078 raise Exception('You can not use a reweighted observable for reweighting!') -1079 -1080 proj_qtop = [] -1081 for n in qtop.deltas: -1082 proj_qtop.append(np.array([1 if round(qtop.r_values[n] + q) == target else 0 for q in qtop.deltas[n]])) +1065 +1066def qtop_projection(qtop, target=0): +1067 """Returns the projection to the topological charge sector defined by target. +1068 +1069 Parameters +1070 ---------- +1071 path : Obs +1072 Topological charge. +1073 target : int +1074 Specifies the topological sector to be reweighted to (default 0) +1075 +1076 Returns +1077 ------- +1078 reto : Obs +1079 projection to the topological charge sector defined by target +1080 """ +1081 if qtop.reweighted: +1082 raise Exception('You can not use a reweighted observable for reweighting!') 1083 -1084 reto = Obs(proj_qtop, qtop.names, idl=[qtop.idl[name] for name in qtop.names]) -1085 return reto -1086 +1084 proj_qtop = [] +1085 for n in qtop.deltas: +1086 proj_qtop.append(np.array([1 if round(qtop.r_values[n] + q) == target else 0 for q in qtop.deltas[n]])) 1087 -1088def read_qtop_sector(path, prefix, c, target=0, **kwargs): -1089 """Constructs reweighting factors to a specified topological sector. +1088 reto = Obs(proj_qtop, qtop.names, idl=[qtop.idl[name] for name in qtop.names]) +1089 return reto 1090 -1091 Parameters -1092 ---------- -1093 path : str -1094 path of the measurement files -1095 prefix : str -1096 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat -1097 c : double -1098 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L -1099 target : int -1100 Specifies the topological sector to be reweighted to (default 0) -1101 dtr_cnfg : int -1102 (optional) parameter that specifies the number of trajectories -1103 between two configs. -1104 if it is not set, the distance between two measurements -1105 in the file is assumed to be the distance between two configurations. -1106 steps : int -1107 (optional) Distance between two configurations in units of trajectories / -1108 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given -1109 version : str -1110 version string of the openQCD (sfqcd) version used to create -1111 the ensemble. Default is 2.0. May also be set to sfqcd. -1112 L : int -1113 spatial length of the lattice in L/a. -1114 HAS to be set if version != sfqcd, since openQCD does not provide -1115 this in the header -1116 r_start : list -1117 offset of the first ensemble, making it easier to match -1118 later on with other Obs -1119 r_stop : list -1120 last configurations that need to be read (per replicum) -1121 files : list -1122 specify the exact files that need to be read -1123 from path, practical if e.g. only one replicum is needed -1124 names : list -1125 Alternative labeling for replicas/ensembles. -1126 Has to have the appropriate length -1127 Zeuthen_flow : bool -1128 (optional) If True, the Zeuthen flow is used for Qtop. Only possible -1129 for version=='sfqcd' If False, the Wilson flow is used. -1130 -1131 Returns -1132 ------- -1133 reto : Obs -1134 projection to the topological charge sector defined by target -1135 """ -1136 -1137 if not isinstance(target, int): -1138 raise Exception("'target' has to be an integer.") -1139 -1140 kwargs['integer_charge'] = True -1141 qtop = read_qtop(path, prefix, c, **kwargs) -1142 -1143 return qtop_projection(qtop, target=target) -1144 -1145 -1146def read_ms5_xsf(path, prefix, qc, corr, sep="r", **kwargs): -1147 """ -1148 Read data from files in the specified directory with the specified prefix and quark combination extension, and return a `Corr` object containing the data. +1091 +1092def read_qtop_sector(path, prefix, c, target=0, **kwargs): +1093 """Constructs reweighting factors to a specified topological sector. +1094 +1095 Parameters +1096 ---------- +1097 path : str +1098 path of the measurement files +1099 prefix : str +1100 prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat +1101 c : double +1102 Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L +1103 target : int +1104 Specifies the topological sector to be reweighted to (default 0) +1105 dtr_cnfg : int +1106 (optional) parameter that specifies the number of trajectories +1107 between two configs. +1108 if it is not set, the distance between two measurements +1109 in the file is assumed to be the distance between two configurations. +1110 steps : int +1111 (optional) Distance between two configurations in units of trajectories / +1112 cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given +1113 version : str +1114 version string of the openQCD (sfqcd) version used to create +1115 the ensemble. Default is 2.0. May also be set to sfqcd. +1116 L : int +1117 spatial length of the lattice in L/a. +1118 HAS to be set if version != sfqcd, since openQCD does not provide +1119 this in the header +1120 r_start : list +1121 offset of the first ensemble, making it easier to match +1122 later on with other Obs +1123 r_stop : list +1124 last configurations that need to be read (per replicum) +1125 files : list +1126 specify the exact files that need to be read +1127 from path, practical if e.g. only one replicum is needed +1128 names : list +1129 Alternative labeling for replicas/ensembles. +1130 Has to have the appropriate length +1131 Zeuthen_flow : bool +1132 (optional) If True, the Zeuthen flow is used for Qtop. Only possible +1133 for version=='sfqcd' If False, the Wilson flow is used. +1134 +1135 Returns +1136 ------- +1137 reto : Obs +1138 projection to the topological charge sector defined by target +1139 """ +1140 +1141 if not isinstance(target, int): +1142 raise Exception("'target' has to be an integer.") +1143 +1144 kwargs['integer_charge'] = True +1145 qtop = read_qtop(path, prefix, c, **kwargs) +1146 +1147 return qtop_projection(qtop, target=target) +1148 1149 -1150 Parameters -1151 ---------- -1152 path : str -1153 The directory to search for the files in. -1154 prefix : str -1155 The prefix to match the files against. -1156 qc : str -1157 The quark combination extension to match the files against. -1158 corr : str -1159 The correlator to extract data for. -1160 sep : str, optional -1161 The separator to use when parsing the replika names. -1162 **kwargs -1163 Additional keyword arguments. The following keyword arguments are recognized: -1164 -1165 - names (List[str]): A list of names to use for the replicas. -1166 - files (List[str]): A list of files to read data from. -1167 - idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given. +1150def read_ms5_xsf(path, prefix, qc, corr, sep="r", **kwargs): +1151 """ +1152 Read data from files in the specified directory with the specified prefix and quark combination extension, and return a `Corr` object containing the data. +1153 +1154 Parameters +1155 ---------- +1156 path : str +1157 The directory to search for the files in. +1158 prefix : str +1159 The prefix to match the files against. +1160 qc : str +1161 The quark combination extension to match the files against. +1162 corr : str +1163 The correlator to extract data for. +1164 sep : str, optional +1165 The separator to use when parsing the replika names. +1166 **kwargs +1167 Additional keyword arguments. The following keyword arguments are recognized: 1168 -1169 Returns -1170 ------- -1171 Corr -1172 A complex valued `Corr` object containing the data read from the files. In case of boudary to bulk correlators. -1173 or -1174 CObs -1175 A complex valued `CObs` object containing the data read from the files. In case of boudary to boundary correlators. -1176 -1177 -1178 Raises -1179 ------ -1180 FileNotFoundError -1181 If no files matching the specified prefix and quark combination extension are found in the specified directory. -1182 IOError -1183 If there is an error reading a file. -1184 struct.error -1185 If there is an error unpacking binary data. -1186 """ -1187 -1188 # found = [] -1189 files = [] -1190 names = [] +1169 - names (List[str]): A list of names to use for the replicas. +1170 - files (List[str]): A list of files to read data from. +1171 - idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given. +1172 +1173 Returns +1174 ------- +1175 Corr +1176 A complex valued `Corr` object containing the data read from the files. In case of boudary to bulk correlators. +1177 or +1178 CObs +1179 A complex valued `CObs` object containing the data read from the files. In case of boudary to boundary correlators. +1180 +1181 +1182 Raises +1183 ------ +1184 FileNotFoundError +1185 If no files matching the specified prefix and quark combination extension are found in the specified directory. +1186 IOError +1187 If there is an error reading a file. +1188 struct.error +1189 If there is an error unpacking binary data. +1190 """ 1191 -1192 # test if the input is correct -1193 if qc not in ['dd', 'ud', 'du', 'uu']: -1194 raise Exception("Unknown quark conbination!") +1192 # found = [] +1193 files = [] +1194 names = [] 1195 -1196 if corr not in ["gS", "gP", "gA", "gV", "gVt", "lA", "lV", "lVt", "lT", "lTt", "g1", "l1"]: -1197 raise Exception("Unknown correlator!") -1198 -1199 if "files" in kwargs: -1200 known_files = kwargs.get("files") -1201 else: -1202 known_files = [] -1203 files = _find_files(path, prefix, "ms5_xsf_" + qc, "dat", known_files=known_files) -1204 -1205 if "names" in kwargs: -1206 names = kwargs.get("names") -1207 else: -1208 for f in files: -1209 if not sep == "": -1210 se = f.split(".")[0] -1211 for s in f.split(".")[1:-2]: -1212 se += "." + s -1213 names.append(se.split(sep)[0] + "|r" + se.split(sep)[1]) -1214 else: -1215 names.append(prefix) -1216 if 'idl' in kwargs: -1217 expected_idl = kwargs.get('idl') -1218 names = sorted(names) -1219 files = sorted(files) -1220 -1221 cnfgs = [] -1222 realsamples = [] -1223 imagsamples = [] -1224 repnum = 0 -1225 for file in files: -1226 with open(path + "/" + file, "rb") as fp: -1227 -1228 t = fp.read(8) -1229 kappa = struct.unpack('d', t)[0] -1230 t = fp.read(8) -1231 csw = struct.unpack('d', t)[0] +1196 # test if the input is correct +1197 if qc not in ['dd', 'ud', 'du', 'uu']: +1198 raise Exception("Unknown quark conbination!") +1199 +1200 if corr not in ["gS", "gP", "gA", "gV", "gVt", "lA", "lV", "lVt", "lT", "lTt", "g1", "l1"]: +1201 raise Exception("Unknown correlator!") +1202 +1203 if "files" in kwargs: +1204 known_files = kwargs.get("files") +1205 else: +1206 known_files = [] +1207 files = _find_files(path, prefix, "ms5_xsf_" + qc, "dat", known_files=known_files) +1208 +1209 if "names" in kwargs: +1210 names = kwargs.get("names") +1211 else: +1212 for f in files: +1213 if not sep == "": +1214 se = f.split(".")[0] +1215 for s in f.split(".")[1:-2]: +1216 se += "." + s +1217 names.append(se.split(sep)[0] + "|r" + se.split(sep)[1]) +1218 else: +1219 names.append(prefix) +1220 if 'idl' in kwargs: +1221 expected_idl = kwargs.get('idl') +1222 names = sorted(names) +1223 files = sorted(files) +1224 +1225 cnfgs = [] +1226 realsamples = [] +1227 imagsamples = [] +1228 repnum = 0 +1229 for file in files: +1230 with open(path + "/" + file, "rb") as fp: +1231 1232 t = fp.read(8) -1233 dF = struct.unpack('d', t)[0] +1233 kappa = struct.unpack('d', t)[0] 1234 t = fp.read(8) -1235 zF = struct.unpack('d', t)[0] -1236 -1237 t = fp.read(4) -1238 tmax = struct.unpack('i', t)[0] -1239 t = fp.read(4) -1240 bnd = struct.unpack('i', t)[0] -1241 -1242 placesBI = ["gS", "gP", -1243 "gA", "gV", -1244 "gVt", "lA", -1245 "lV", "lVt", -1246 "lT", "lTt"] -1247 placesBB = ["g1", "l1"] -1248 -1249 # the chunks have the following structure: -1250 # confignumber, 10x timedependent complex correlators as doubles, 2x timeindependent complex correlators as doubles -1251 -1252 chunksize = 4 + (8 * 2 * tmax * 10) + (8 * 2 * 2) -1253 packstr = '=i' + ('d' * 2 * tmax * 10) + ('d' * 2 * 2) -1254 cnfgs.append([]) -1255 realsamples.append([]) -1256 imagsamples.append([]) -1257 for t in range(tmax): -1258 realsamples[repnum].append([]) -1259 imagsamples[repnum].append([]) -1260 if 'idl' in kwargs: -1261 left_idl = set(expected_idl[repnum]) -1262 while True: -1263 cnfgt = fp.read(chunksize) -1264 if not cnfgt: -1265 break -1266 asascii = struct.unpack(packstr, cnfgt) -1267 cnfg = asascii[0] -1268 idl_wanted = True -1269 if 'idl' in kwargs: -1270 idl_wanted = (cnfg in expected_idl[repnum]) -1271 left_idl = left_idl - set([cnfg]) -1272 if idl_wanted: -1273 cnfgs[repnum].append(cnfg) -1274 -1275 if corr not in placesBB: -1276 tmpcorr = asascii[1 + 2 * tmax * placesBI.index(corr):1 + 2 * tmax * placesBI.index(corr) + 2 * tmax] -1277 else: -1278 tmpcorr = asascii[1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr):1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr) + 2] -1279 -1280 corrres = [[], []] -1281 for i in range(len(tmpcorr)): -1282 corrres[i % 2].append(tmpcorr[i]) -1283 for t in range(int(len(tmpcorr) / 2)): -1284 realsamples[repnum][t].append(corrres[0][t]) -1285 for t in range(int(len(tmpcorr) / 2)): -1286 imagsamples[repnum][t].append(corrres[1][t]) -1287 if 'idl' in kwargs: -1288 left_idl = list(left_idl) -1289 if expected_idl[repnum] == left_idl: -1290 raise ValueError("None of the idls searched for were found in replikum of file " + file) -1291 elif len(left_idl) > 0: -1292 warnings.warn('Could not find idls ' + str(left_idl) + ' in replikum of file ' + file, UserWarning) -1293 repnum += 1 -1294 s = "Read correlator " + corr + " from " + str(repnum) + " replika with idls" + str(realsamples[0][t]) -1295 for rep in range(1, repnum): -1296 s += ", " + str(realsamples[rep][t]) -1297 print(s) -1298 print("Asserted run parameters:\n T:", tmax, "kappa:", kappa, "csw:", csw, "dF:", dF, "zF:", zF, "bnd:", bnd) -1299 -1300 # we have the data now... but we need to re format the whole thing and put it into Corr objects. -1301 -1302 compObs = [] +1235 csw = struct.unpack('d', t)[0] +1236 t = fp.read(8) +1237 dF = struct.unpack('d', t)[0] +1238 t = fp.read(8) +1239 zF = struct.unpack('d', t)[0] +1240 +1241 t = fp.read(4) +1242 tmax = struct.unpack('i', t)[0] +1243 t = fp.read(4) +1244 bnd = struct.unpack('i', t)[0] +1245 +1246 placesBI = ["gS", "gP", +1247 "gA", "gV", +1248 "gVt", "lA", +1249 "lV", "lVt", +1250 "lT", "lTt"] +1251 placesBB = ["g1", "l1"] +1252 +1253 # the chunks have the following structure: +1254 # confignumber, 10x timedependent complex correlators as doubles, 2x timeindependent complex correlators as doubles +1255 +1256 chunksize = 4 + (8 * 2 * tmax * 10) + (8 * 2 * 2) +1257 packstr = '=i' + ('d' * 2 * tmax * 10) + ('d' * 2 * 2) +1258 cnfgs.append([]) +1259 realsamples.append([]) +1260 imagsamples.append([]) +1261 for t in range(tmax): +1262 realsamples[repnum].append([]) +1263 imagsamples[repnum].append([]) +1264 if 'idl' in kwargs: +1265 left_idl = set(expected_idl[repnum]) +1266 while True: +1267 cnfgt = fp.read(chunksize) +1268 if not cnfgt: +1269 break +1270 asascii = struct.unpack(packstr, cnfgt) +1271 cnfg = asascii[0] +1272 idl_wanted = True +1273 if 'idl' in kwargs: +1274 idl_wanted = (cnfg in expected_idl[repnum]) +1275 left_idl = left_idl - set([cnfg]) +1276 if idl_wanted: +1277 cnfgs[repnum].append(cnfg) +1278 +1279 if corr not in placesBB: +1280 tmpcorr = asascii[1 + 2 * tmax * placesBI.index(corr):1 + 2 * tmax * placesBI.index(corr) + 2 * tmax] +1281 else: +1282 tmpcorr = asascii[1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr):1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr) + 2] +1283 +1284 corrres = [[], []] +1285 for i in range(len(tmpcorr)): +1286 corrres[i % 2].append(tmpcorr[i]) +1287 for t in range(int(len(tmpcorr) / 2)): +1288 realsamples[repnum][t].append(corrres[0][t]) +1289 for t in range(int(len(tmpcorr) / 2)): +1290 imagsamples[repnum][t].append(corrres[1][t]) +1291 if 'idl' in kwargs: +1292 left_idl = list(left_idl) +1293 if expected_idl[repnum] == left_idl: +1294 raise ValueError("None of the idls searched for were found in replikum of file " + file) +1295 elif len(left_idl) > 0: +1296 warnings.warn('Could not find idls ' + str(left_idl) + ' in replikum of file ' + file, UserWarning) +1297 repnum += 1 +1298 s = "Read correlator " + corr + " from " + str(repnum) + " replika with idls" + str(realsamples[0][t]) +1299 for rep in range(1, repnum): +1300 s += ", " + str(realsamples[rep][t]) +1301 print(s) +1302 print("Asserted run parameters:\n T:", tmax, "kappa:", kappa, "csw:", csw, "dF:", dF, "zF:", zF, "bnd:", bnd) 1303 -1304 for t in range(int(len(tmpcorr) / 2)): -1305 compObs.append(CObs(Obs([realsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs), -1306 Obs([imagsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs))) +1304 # we have the data now... but we need to re format the whole thing and put it into Corr objects. +1305 +1306 compObs = [] 1307 -1308 if len(compObs) == 1: -1309 return compObs[0] -1310 else: -1311 return Corr(compObs) +1308 for t in range(int(len(tmpcorr) / 2)): +1309 compObs.append(CObs(Obs([realsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs), +1310 Obs([imagsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs))) +1311 +1312 if len(compObs) == 1: +1313 return compObs[0] +1314 else: +1315 return Corr(compObs) @@ -1696,77 +1700,78 @@ Reweighting factors read - 
426def extract_t0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs):
-427    """Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
-428
-429    It is assumed that all boundary effects have
-430    sufficiently decayed at x0=xmin.
-431    The data around the zero crossing of t^2<E> - c (where c=0.3 by default)
-432    is fitted with a linear function
-433    from which the exact root is extracted.
-434
-435    It is assumed that one measurement is performed for each config.
-436    If this is not the case, the resulting idl, as well as the handling
-437    of r_start, r_stop and r_step is wrong and the user has to correct
-438    this in the resulting observable.
-439
-440    Parameters
-441    ----------
-442    path : str
-443        Path to .ms.dat files
-444    prefix : str
-445        Ensemble prefix
-446    dtr_read : int
-447        Determines how many trajectories should be skipped
-448        when reading the ms.dat files.
-449        Corresponds to dtr_cnfg / dtr_ms in the openQCD input file.
-450    xmin : int
-451        First timeslice where the boundary
-452        effects have sufficiently decayed.
-453    spatial_extent : int
-454        spatial extent of the lattice, required for normalization.
-455    fit_range : int
-456        Number of data points left and right of the zero
-457        crossing to be included in the linear fit. (Default: 5)
-458    postfix : str
-459        Postfix of measurement file (Default: ms)
-460    c: float
-461        Constant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
-462    r_start : list
-463        list which contains the first config to be read for each replicum.
-464    r_stop : list
-465        list which contains the last config to be read for each replicum.
-466    r_step : int
-467        integer that defines a fixed step size between two measurements (in units of configs)
-468        If not given, r_step=1 is assumed.
-469    plaquette : bool
-470        If true extract the plaquette estimate of t0 instead.
-471    names : list
-472        list of names that is assigned to the data according according
-473        to the order in the file list. Use careful, if you do not provide file names!
-474    files : list
-475        list which contains the filenames to be read. No automatic detection of
-476        files performed if given.
-477    plot_fit : bool
-478        If true, the fit for the extraction of t0 is shown together with the data.
-479    assume_thermalization : bool
-480        If True: If the first record divided by the distance between two measurements is larger than
-481        1, it is assumed that this is due to thermalization and the first measurement belongs
-482        to the first config (default).
-483        If False: The config numbers are assumed to be traj_number // difference
-484
-485    Returns
-486    -------
-487    t0 : Obs
-488        Extracted t0
-489    """
-490
-491    E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs)
-492    t2E_dict = {}
-493    for t in sorted(E_dict.keys()):
-494        t2E_dict[t] = t ** 2 * E_dict[t] - c
-495
-496    return fit_t0(t2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'))
+            
429def extract_t0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs):
+430    """Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
+431
+432    It is assumed that all boundary effects have
+433    sufficiently decayed at x0=xmin.
+434    The data around the zero crossing of t^2<E> - c (where c=0.3 by default)
+435    is fitted with a linear function
+436    from which the exact root is extracted.
+437
+438    It is assumed that one measurement is performed for each config.
+439    If this is not the case, the resulting idl, as well as the handling
+440    of `r_start`, `r_stop` and `r_step` is wrong and the user has to correct
+441    this in the resulting observable.
+442    The function also assumes that `r_step` is the same across all replica.
+443
+444    Parameters
+445    ----------
+446    path : str
+447        Path to .ms.dat files
+448    prefix : str
+449        Ensemble prefix
+450    dtr_read : int
+451        Determines how many trajectories should be skipped
+452        when reading the ms.dat files.
+453        Corresponds to dtr_cnfg / dtr_ms in the openQCD input file.
+454    xmin : int
+455        First timeslice where the boundary
+456        effects have sufficiently decayed.
+457    spatial_extent : int
+458        spatial extent of the lattice, required for normalization.
+459    fit_range : int
+460        Number of data points left and right of the zero
+461        crossing to be included in the linear fit. (Default: 5)
+462    postfix : str
+463        Postfix of measurement file (Default: ms)
+464    c: float
+465        Constant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
+466    r_start : list
+467        list which contains the first config to be read for each replicum.
+468    r_stop : list
+469        list which contains the last config to be read for each replicum.
+470    r_step : int
+471        integer that defines a fixed step size between two measurements (in units of configs)
+472        If not given, r_step=1 is assumed.
+473    plaquette : bool
+474        If true extract the plaquette estimate of t0 instead.
+475    names : list
+476        list of names that is assigned to the data according according
+477        to the order in the file list. Use careful, if you do not provide file names!
+478    files : list
+479        list which contains the filenames to be read. No automatic detection of
+480        files performed if given.
+481    plot_fit : bool
+482        If true, the fit for the extraction of t0 is shown together with the data.
+483    assume_thermalization : bool
+484        If True: If the first record divided by the distance between two measurements is larger than
+485        1, it is assumed that this is due to thermalization and the first measurement belongs
+486        to the first config (default).
+487        If False: The config numbers are assumed to be traj_number // difference
+488
+489    Returns
+490    -------
+491    t0 : Obs
+492        Extracted t0
+493    """
+494
+495    E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs)
+496    t2E_dict = {}
+497    for t in sorted(E_dict.keys()):
+498        t2E_dict[t] = t ** 2 * E_dict[t] - c
+499
+500    return fit_t0(t2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'))
 

 
 
@@ -1780,8 +1785,9 @@ from which the exact root is extracted.

 
 
It is assumed that one measurement is performed for each config.
 If this is not the case, the resulting idl, as well as the handling
-of r_start, r_stop and r_step is wrong and the user has to correct
-this in the resulting observable.

+of r_start, r_stop and r_step is wrong and the user has to correct
+this in the resulting observable.
+The function also assumes that r_step is the same across all replica.

 
 
Parameters

 
@@ -1851,86 +1857,86 @@ Extracted t0
- 
499def extract_w0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs):
-500    """Extract w0/a from given .ms.dat files. Returns w0 as Obs.
-501
-502    It is assumed that all boundary effects have
-503    sufficiently decayed at x0=xmin.
-504    The data around the zero crossing of t d(t^2<E>)/dt -  (where c=0.3 by default)
-505    is fitted with a linear function
-506    from which the exact root is extracted.
-507
-508    It is assumed that one measurement is performed for each config.
-509    If this is not the case, the resulting idl, as well as the handling
-510    of r_start, r_stop and r_step is wrong and the user has to correct
-511    this in the resulting observable.
-512
-513    Parameters
-514    ----------
-515    path : str
-516        Path to .ms.dat files
-517    prefix : str
-518        Ensemble prefix
-519    dtr_read : int
-520        Determines how many trajectories should be skipped
-521        when reading the ms.dat files.
-522        Corresponds to dtr_cnfg / dtr_ms in the openQCD input file.
-523    xmin : int
-524        First timeslice where the boundary
-525        effects have sufficiently decayed.
-526    spatial_extent : int
-527        spatial extent of the lattice, required for normalization.
-528    fit_range : int
-529        Number of data points left and right of the zero
-530        crossing to be included in the linear fit. (Default: 5)
-531    postfix : str
-532        Postfix of measurement file (Default: ms)
-533    c: float
-534        Constant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
-535    r_start : list
-536        list which contains the first config to be read for each replicum.
-537    r_stop : list
-538        list which contains the last config to be read for each replicum.
-539    r_step : int
-540        integer that defines a fixed step size between two measurements (in units of configs)
-541        If not given, r_step=1 is assumed.
-542    plaquette : bool
-543        If true extract the plaquette estimate of w0 instead.
-544    names : list
-545        list of names that is assigned to the data according according
-546        to the order in the file list. Use careful, if you do not provide file names!
-547    files : list
-548        list which contains the filenames to be read. No automatic detection of
-549        files performed if given.
-550    plot_fit : bool
-551        If true, the fit for the extraction of w0 is shown together with the data.
-552    assume_thermalization : bool
-553        If True: If the first record divided by the distance between two measurements is larger than
-554        1, it is assumed that this is due to thermalization and the first measurement belongs
-555        to the first config (default).
-556        If False: The config numbers are assumed to be traj_number // difference
-557
-558    Returns
-559    -------
-560    w0 : Obs
-561        Extracted w0
-562    """
-563
-564    E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs)
-565
-566    ftimes = sorted(E_dict.keys())
+            
503def extract_w0(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, postfix='ms', c=0.3, **kwargs):
+504    """Extract w0/a from given .ms.dat files. Returns w0 as Obs.
+505
+506    It is assumed that all boundary effects have
+507    sufficiently decayed at x0=xmin.
+508    The data around the zero crossing of t d(t^2<E>)/dt -  (where c=0.3 by default)
+509    is fitted with a linear function
+510    from which the exact root is extracted.
+511
+512    It is assumed that one measurement is performed for each config.
+513    If this is not the case, the resulting idl, as well as the handling
+514    of r_start, r_stop and r_step is wrong and the user has to correct
+515    this in the resulting observable.
+516
+517    Parameters
+518    ----------
+519    path : str
+520        Path to .ms.dat files
+521    prefix : str
+522        Ensemble prefix
+523    dtr_read : int
+524        Determines how many trajectories should be skipped
+525        when reading the ms.dat files.
+526        Corresponds to dtr_cnfg / dtr_ms in the openQCD input file.
+527    xmin : int
+528        First timeslice where the boundary
+529        effects have sufficiently decayed.
+530    spatial_extent : int
+531        spatial extent of the lattice, required for normalization.
+532    fit_range : int
+533        Number of data points left and right of the zero
+534        crossing to be included in the linear fit. (Default: 5)
+535    postfix : str
+536        Postfix of measurement file (Default: ms)
+537    c: float
+538        Constant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
+539    r_start : list
+540        list which contains the first config to be read for each replicum.
+541    r_stop : list
+542        list which contains the last config to be read for each replicum.
+543    r_step : int
+544        integer that defines a fixed step size between two measurements (in units of configs)
+545        If not given, r_step=1 is assumed.
+546    plaquette : bool
+547        If true extract the plaquette estimate of w0 instead.
+548    names : list
+549        list of names that is assigned to the data according according
+550        to the order in the file list. Use careful, if you do not provide file names!
+551    files : list
+552        list which contains the filenames to be read. No automatic detection of
+553        files performed if given.
+554    plot_fit : bool
+555        If true, the fit for the extraction of w0 is shown together with the data.
+556    assume_thermalization : bool
+557        If True: If the first record divided by the distance between two measurements is larger than
+558        1, it is assumed that this is due to thermalization and the first measurement belongs
+559        to the first config (default).
+560        If False: The config numbers are assumed to be traj_number // difference
+561
+562    Returns
+563    -------
+564    w0 : Obs
+565        Extracted w0
+566    """
 567
-568    t2E_dict = {}
-569    for t in ftimes:
-570        t2E_dict[t] = t ** 2 * E_dict[t]
+568    E_dict = _extract_flowed_energy_density(path, prefix, dtr_read, xmin, spatial_extent, postfix, **kwargs)
+569
+570    ftimes = sorted(E_dict.keys())
 571
-572    tdtt2E_dict = {}
-573    tdtt2E_dict[ftimes[0]] = ftimes[0] * (t2E_dict[ftimes[1]] - t2E_dict[ftimes[0]]) / (ftimes[1] - ftimes[0]) - c
-574    for i in range(1, len(ftimes) - 1):
-575        tdtt2E_dict[ftimes[i]] = ftimes[i] * (t2E_dict[ftimes[i + 1]] - t2E_dict[ftimes[i - 1]]) / (ftimes[i + 1] - ftimes[i - 1]) - c
-576    tdtt2E_dict[ftimes[-1]] = ftimes[-1] * (t2E_dict[ftimes[-1]] - t2E_dict[ftimes[-2]]) / (ftimes[-1] - ftimes[-2]) - c
-577
-578    return np.sqrt(fit_t0(tdtt2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'), observable='w0'))
+572    t2E_dict = {}
+573    for t in ftimes:
+574        t2E_dict[t] = t ** 2 * E_dict[t]
+575
+576    tdtt2E_dict = {}
+577    tdtt2E_dict[ftimes[0]] = ftimes[0] * (t2E_dict[ftimes[1]] - t2E_dict[ftimes[0]]) / (ftimes[1] - ftimes[0]) - c
+578    for i in range(1, len(ftimes) - 1):
+579        tdtt2E_dict[ftimes[i]] = ftimes[i] * (t2E_dict[ftimes[i + 1]] - t2E_dict[ftimes[i - 1]]) / (ftimes[i + 1] - ftimes[i - 1]) - c
+580    tdtt2E_dict[ftimes[-1]] = ftimes[-1] * (t2E_dict[ftimes[-1]] - t2E_dict[ftimes[-2]]) / (ftimes[-1] - ftimes[-2]) - c
+581
+582    return np.sqrt(fit_t0(tdtt2E_dict, fit_range, plot_fit=kwargs.get('plot_fit'), observable='w0'))
 

 
 
@@ -2015,57 +2021,57 @@ Extracted w0
- 
666def read_qtop(path, prefix, c, dtr_cnfg=1, version="openQCD", **kwargs):
-667    """Read the topologial charge based on openQCD gradient flow measurements.
-668
-669    Parameters
-670    ----------
-671    path : str
-672        path of the measurement files
-673    prefix : str
-674        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat.
-675        Ignored if file names are passed explicitly via keyword files.
-676    c : double
-677        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
-678    dtr_cnfg : int
-679        (optional) parameter that specifies the number of measurements
-680        between two configs.
-681        If it is not set, the distance between two measurements
-682        in the file is assumed to be the distance between two configurations.
-683    steps : int
-684        (optional) Distance between two configurations in units of trajectories /
-685         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
-686    version : str
-687        Either openQCD or sfqcd, depending on the data.
-688    L : int
-689        spatial length of the lattice in L/a.
-690        HAS to be set if version != sfqcd, since openQCD does not provide
-691        this in the header
-692    r_start : list
-693        list which contains the first config to be read for each replicum.
-694    r_stop : list
-695        list which contains the last config to be read for each replicum.
-696    files : list
-697        specify the exact files that need to be read
-698        from path, practical if e.g. only one replicum is needed
-699    postfix : str
-700        postfix of the file to read, e.g. '.gfms.dat' for openQCD-files
-701    names : list
-702        Alternative labeling for replicas/ensembles.
-703        Has to have the appropriate length.
-704    Zeuthen_flow : bool
-705        (optional) If True, the Zeuthen flow is used for Qtop. Only possible
-706        for version=='sfqcd' If False, the Wilson flow is used.
-707    integer_charge : bool
-708        If True, the charge is rounded towards the nearest integer on each config.
-709
-710    Returns
-711    -------
-712    result : Obs
-713        Read topological charge
-714    """
-715
-716    return _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version=version, obspos=0, **kwargs)
+            
670def read_qtop(path, prefix, c, dtr_cnfg=1, version="openQCD", **kwargs):
+671    """Read the topologial charge based on openQCD gradient flow measurements.
+672
+673    Parameters
+674    ----------
+675    path : str
+676        path of the measurement files
+677    prefix : str
+678        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat.
+679        Ignored if file names are passed explicitly via keyword files.
+680    c : double
+681        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
+682    dtr_cnfg : int
+683        (optional) parameter that specifies the number of measurements
+684        between two configs.
+685        If it is not set, the distance between two measurements
+686        in the file is assumed to be the distance between two configurations.
+687    steps : int
+688        (optional) Distance between two configurations in units of trajectories /
+689         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
+690    version : str
+691        Either openQCD or sfqcd, depending on the data.
+692    L : int
+693        spatial length of the lattice in L/a.
+694        HAS to be set if version != sfqcd, since openQCD does not provide
+695        this in the header
+696    r_start : list
+697        list which contains the first config to be read for each replicum.
+698    r_stop : list
+699        list which contains the last config to be read for each replicum.
+700    files : list
+701        specify the exact files that need to be read
+702        from path, practical if e.g. only one replicum is needed
+703    postfix : str
+704        postfix of the file to read, e.g. '.gfms.dat' for openQCD-files
+705    names : list
+706        Alternative labeling for replicas/ensembles.
+707        Has to have the appropriate length.
+708    Zeuthen_flow : bool
+709        (optional) If True, the Zeuthen flow is used for Qtop. Only possible
+710        for version=='sfqcd' If False, the Wilson flow is used.
+711    integer_charge : bool
+712        If True, the charge is rounded towards the nearest integer on each config.
+713
+714    Returns
+715    -------
+716    result : Obs
+717        Read topological charge
+718    """
+719
+720    return _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version=version, obspos=0, **kwargs)
 

 
 
@@ -2135,76 +2141,76 @@ Read topological charge
- 
719def read_gf_coupling(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):
-720    """Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
-721
-722    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
-723
-724    Parameters
-725    ----------
-726    path : str
-727        path of the measurement files
-728    prefix : str
-729        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat.
-730        Ignored if file names are passed explicitly via keyword files.
-731    c : double
-732        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
-733    dtr_cnfg : int
-734        (optional) parameter that specifies the number of measurements
-735        between two configs.
-736        If it is not set, the distance between two measurements
-737        in the file is assumed to be the distance between two configurations.
-738    steps : int
-739        (optional) Distance between two configurations in units of trajectories /
-740         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
-741    r_start : list
-742        list which contains the first config to be read for each replicum.
-743    r_stop : list
-744        list which contains the last config to be read for each replicum.
-745    files : list
-746        specify the exact files that need to be read
-747        from path, practical if e.g. only one replicum is needed
-748    names : list
-749        Alternative labeling for replicas/ensembles.
-750        Has to have the appropriate length.
-751    postfix : str
-752        postfix of the file to read, e.g. '.gfms.dat' for openQCD-files
-753    Zeuthen_flow : bool
-754        (optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
-755    """
-756
-757    if c != 0.3:
-758        raise Exception("The required lattice norm is only implemented for c=0.3 at the moment.")
-759
-760    plaq = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=6, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs)
-761    C2x1 = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=7, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs)
-762    L = plaq.tag["L"]
-763    T = plaq.tag["T"]
-764
-765    if T != L:
-766        raise Exception("The required lattice norm is only implemented for T=L at the moment.")
-767
-768    if Zeuthen_flow is not True:
-769        raise Exception("The required lattice norm is only implemented for the Zeuthen flow at the moment.")
-770
-771    t = (c * L) ** 2 / 8
-772
-773    normdict = {4: 0.012341170468270,
-774                6: 0.010162691462430,
-775                8: 0.009031614807931,
-776                10: 0.008744966371393,
-777                12: 0.008650917856809,
-778                14: 8.611154391267955E-03,
-779                16: 0.008591758449508,
-780                20: 0.008575359627103,
-781                24: 0.008569387847540,
-782                28: 8.566803713382559E-03,
-783                32: 0.008565541650006,
-784                40: 8.564480684962046E-03,
-785                48: 8.564098025073460E-03,
-786                64: 8.563853943383087E-03}
-787
-788    return t * t * (5 / 3 * plaq - 1 / 12 * C2x1) / normdict[L]
+            
723def read_gf_coupling(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):
+724    """Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
+725
+726    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
+727
+728    Parameters
+729    ----------
+730    path : str
+731        path of the measurement files
+732    prefix : str
+733        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat.
+734        Ignored if file names are passed explicitly via keyword files.
+735    c : double
+736        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
+737    dtr_cnfg : int
+738        (optional) parameter that specifies the number of measurements
+739        between two configs.
+740        If it is not set, the distance between two measurements
+741        in the file is assumed to be the distance between two configurations.
+742    steps : int
+743        (optional) Distance between two configurations in units of trajectories /
+744         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
+745    r_start : list
+746        list which contains the first config to be read for each replicum.
+747    r_stop : list
+748        list which contains the last config to be read for each replicum.
+749    files : list
+750        specify the exact files that need to be read
+751        from path, practical if e.g. only one replicum is needed
+752    names : list
+753        Alternative labeling for replicas/ensembles.
+754        Has to have the appropriate length.
+755    postfix : str
+756        postfix of the file to read, e.g. '.gfms.dat' for openQCD-files
+757    Zeuthen_flow : bool
+758        (optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
+759    """
+760
+761    if c != 0.3:
+762        raise Exception("The required lattice norm is only implemented for c=0.3 at the moment.")
+763
+764    plaq = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=6, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs)
+765    C2x1 = _read_flow_obs(path, prefix, c, dtr_cnfg=dtr_cnfg, version="sfqcd", obspos=7, sum_t=False, Zeuthen_flow=Zeuthen_flow, integer_charge=False, **kwargs)
+766    L = plaq.tag["L"]
+767    T = plaq.tag["T"]
+768
+769    if T != L:
+770        raise Exception("The required lattice norm is only implemented for T=L at the moment.")
+771
+772    if Zeuthen_flow is not True:
+773        raise Exception("The required lattice norm is only implemented for the Zeuthen flow at the moment.")
+774
+775    t = (c * L) ** 2 / 8
+776
+777    normdict = {4: 0.012341170468270,
+778                6: 0.010162691462430,
+779                8: 0.009031614807931,
+780                10: 0.008744966371393,
+781                12: 0.008650917856809,
+782                14: 8.611154391267955E-03,
+783                16: 0.008591758449508,
+784                20: 0.008575359627103,
+785                24: 0.008569387847540,
+786                28: 8.566803713382559E-03,
+787                32: 0.008565541650006,
+788                40: 8.564480684962046E-03,
+789                48: 8.564098025073460E-03,
+790                64: 8.563853943383087E-03}
+791
+792    return t * t * (5 / 3 * plaq - 1 / 12 * C2x1) / normdict[L]
 

 
 
@@ -2260,30 +2266,30 @@ postfix of the file to read, e.g. '.gfms.dat' for openQCD-files
- 
1063def qtop_projection(qtop, target=0):
-1064    """Returns the projection to the topological charge sector defined by target.
-1065
-1066    Parameters
-1067    ----------
-1068    path : Obs
-1069        Topological charge.
-1070    target : int
-1071        Specifies the topological sector to be reweighted to (default 0)
-1072
-1073    Returns
-1074    -------
-1075    reto : Obs
-1076        projection to the topological charge sector defined by target
-1077    """
-1078    if qtop.reweighted:
-1079        raise Exception('You can not use a reweighted observable for reweighting!')
-1080
-1081    proj_qtop = []
-1082    for n in qtop.deltas:
-1083        proj_qtop.append(np.array([1 if round(qtop.r_values[n] + q) == target else 0 for q in qtop.deltas[n]]))
+            
1067def qtop_projection(qtop, target=0):
+1068    """Returns the projection to the topological charge sector defined by target.
+1069
+1070    Parameters
+1071    ----------
+1072    path : Obs
+1073        Topological charge.
+1074    target : int
+1075        Specifies the topological sector to be reweighted to (default 0)
+1076
+1077    Returns
+1078    -------
+1079    reto : Obs
+1080        projection to the topological charge sector defined by target
+1081    """
+1082    if qtop.reweighted:
+1083        raise Exception('You can not use a reweighted observable for reweighting!')
 1084
-1085    reto = Obs(proj_qtop, qtop.names, idl=[qtop.idl[name] for name in qtop.names])
-1086    return reto
+1085    proj_qtop = []
+1086    for n in qtop.deltas:
+1087        proj_qtop.append(np.array([1 if round(qtop.r_values[n] + q) == target else 0 for q in qtop.deltas[n]]))
+1088
+1089    reto = Obs(proj_qtop, qtop.names, idl=[qtop.idl[name] for name in qtop.names])
+1090    return reto
 

 
 
@@ -2319,62 +2325,62 @@ projection to the topological charge sector defined by target
- 
1089def read_qtop_sector(path, prefix, c, target=0, **kwargs):
-1090    """Constructs reweighting factors to a specified topological sector.
-1091
-1092    Parameters
-1093    ----------
-1094    path : str
-1095        path of the measurement files
-1096    prefix : str
-1097        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat
-1098    c : double
-1099        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L
-1100    target : int
-1101        Specifies the topological sector to be reweighted to (default 0)
-1102    dtr_cnfg : int
-1103        (optional) parameter that specifies the number of trajectories
-1104        between two configs.
-1105        if it is not set, the distance between two measurements
-1106        in the file is assumed to be the distance between two configurations.
-1107    steps : int
-1108        (optional) Distance between two configurations in units of trajectories /
-1109         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
-1110    version : str
-1111        version string of the openQCD (sfqcd) version used to create
-1112        the ensemble. Default is 2.0. May also be set to sfqcd.
-1113    L : int
-1114        spatial length of the lattice in L/a.
-1115        HAS to be set if version != sfqcd, since openQCD does not provide
-1116        this in the header
-1117    r_start : list
-1118        offset of the first ensemble, making it easier to match
-1119        later on with other Obs
-1120    r_stop : list
-1121        last configurations that need to be read (per replicum)
-1122    files : list
-1123        specify the exact files that need to be read
-1124        from path, practical if e.g. only one replicum is needed
-1125    names : list
-1126        Alternative labeling for replicas/ensembles.
-1127        Has to have the appropriate length
-1128    Zeuthen_flow : bool
-1129        (optional) If True, the Zeuthen flow is used for Qtop. Only possible
-1130        for version=='sfqcd' If False, the Wilson flow is used.
-1131
-1132    Returns
-1133    -------
-1134    reto : Obs
-1135        projection to the topological charge sector defined by target
-1136    """
-1137
-1138    if not isinstance(target, int):
-1139        raise Exception("'target' has to be an integer.")
-1140
-1141    kwargs['integer_charge'] = True
-1142    qtop = read_qtop(path, prefix, c, **kwargs)
-1143
-1144    return qtop_projection(qtop, target=target)
+            
1093def read_qtop_sector(path, prefix, c, target=0, **kwargs):
+1094    """Constructs reweighting factors to a specified topological sector.
+1095
+1096    Parameters
+1097    ----------
+1098    path : str
+1099        path of the measurement files
+1100    prefix : str
+1101        prefix of the measurement files, e.g. <prefix>_id0_r0.ms.dat
+1102    c : double
+1103        Smearing radius in units of the lattice extent, c = sqrt(8 t0) / L
+1104    target : int
+1105        Specifies the topological sector to be reweighted to (default 0)
+1106    dtr_cnfg : int
+1107        (optional) parameter that specifies the number of trajectories
+1108        between two configs.
+1109        if it is not set, the distance between two measurements
+1110        in the file is assumed to be the distance between two configurations.
+1111    steps : int
+1112        (optional) Distance between two configurations in units of trajectories /
+1113         cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
+1114    version : str
+1115        version string of the openQCD (sfqcd) version used to create
+1116        the ensemble. Default is 2.0. May also be set to sfqcd.
+1117    L : int
+1118        spatial length of the lattice in L/a.
+1119        HAS to be set if version != sfqcd, since openQCD does not provide
+1120        this in the header
+1121    r_start : list
+1122        offset of the first ensemble, making it easier to match
+1123        later on with other Obs
+1124    r_stop : list
+1125        last configurations that need to be read (per replicum)
+1126    files : list
+1127        specify the exact files that need to be read
+1128        from path, practical if e.g. only one replicum is needed
+1129    names : list
+1130        Alternative labeling for replicas/ensembles.
+1131        Has to have the appropriate length
+1132    Zeuthen_flow : bool
+1133        (optional) If True, the Zeuthen flow is used for Qtop. Only possible
+1134        for version=='sfqcd' If False, the Wilson flow is used.
+1135
+1136    Returns
+1137    -------
+1138    reto : Obs
+1139        projection to the topological charge sector defined by target
+1140    """
+1141
+1142    if not isinstance(target, int):
+1143        raise Exception("'target' has to be an integer.")
+1144
+1145    kwargs['integer_charge'] = True
+1146    qtop = read_qtop(path, prefix, c, **kwargs)
+1147
+1148    return qtop_projection(qtop, target=target)
 

 
 
@@ -2443,172 +2449,172 @@ projection to the topological charge sector defined by target
- 
1147def read_ms5_xsf(path, prefix, qc, corr, sep="r", **kwargs):
-1148    """
-1149    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a `Corr` object containing the data.
-1150
-1151    Parameters
-1152    ----------
-1153    path : str
-1154        The directory to search for the files in.
-1155    prefix : str
-1156        The prefix to match the files against.
-1157    qc : str
-1158        The quark combination extension to match the files against.
-1159    corr : str
-1160        The correlator to extract data for.
-1161    sep : str, optional
-1162        The separator to use when parsing the replika names.
-1163    **kwargs
-1164        Additional keyword arguments. The following keyword arguments are recognized:
-1165
-1166        - names (List[str]): A list of names to use for the replicas.
-1167        - files (List[str]): A list of files to read data from.
-1168        - idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
+            
1151def read_ms5_xsf(path, prefix, qc, corr, sep="r", **kwargs):
+1152    """
+1153    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a `Corr` object containing the data.
+1154
+1155    Parameters
+1156    ----------
+1157    path : str
+1158        The directory to search for the files in.
+1159    prefix : str
+1160        The prefix to match the files against.
+1161    qc : str
+1162        The quark combination extension to match the files against.
+1163    corr : str
+1164        The correlator to extract data for.
+1165    sep : str, optional
+1166        The separator to use when parsing the replika names.
+1167    **kwargs
+1168        Additional keyword arguments. The following keyword arguments are recognized:
 1169
-1170    Returns
-1171    -------
-1172    Corr
-1173        A complex valued `Corr` object containing the data read from the files. In case of boudary to bulk correlators.
-1174    or
-1175    CObs
-1176        A complex valued `CObs` object containing the data read from the files. In case of boudary to boundary correlators.
-1177
-1178
-1179    Raises
-1180    ------
-1181    FileNotFoundError
-1182        If no files matching the specified prefix and quark combination extension are found in the specified directory.
-1183    IOError
-1184        If there is an error reading a file.
-1185    struct.error
-1186        If there is an error unpacking binary data.
-1187    """
-1188
-1189    # found = []
-1190    files = []
-1191    names = []
+1170        - names (List[str]): A list of names to use for the replicas.
+1171        - files (List[str]): A list of files to read data from.
+1172        - idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
+1173
+1174    Returns
+1175    -------
+1176    Corr
+1177        A complex valued `Corr` object containing the data read from the files. In case of boudary to bulk correlators.
+1178    or
+1179    CObs
+1180        A complex valued `CObs` object containing the data read from the files. In case of boudary to boundary correlators.
+1181
+1182
+1183    Raises
+1184    ------
+1185    FileNotFoundError
+1186        If no files matching the specified prefix and quark combination extension are found in the specified directory.
+1187    IOError
+1188        If there is an error reading a file.
+1189    struct.error
+1190        If there is an error unpacking binary data.
+1191    """
 1192
-1193    # test if the input is correct
-1194    if qc not in ['dd', 'ud', 'du', 'uu']:
-1195        raise Exception("Unknown quark conbination!")
+1193    # found = []
+1194    files = []
+1195    names = []
 1196
-1197    if corr not in ["gS", "gP", "gA", "gV", "gVt", "lA", "lV", "lVt", "lT", "lTt", "g1", "l1"]:
-1198        raise Exception("Unknown correlator!")
-1199
-1200    if "files" in kwargs:
-1201        known_files = kwargs.get("files")
-1202    else:
-1203        known_files = []
-1204    files = _find_files(path, prefix, "ms5_xsf_" + qc, "dat", known_files=known_files)
-1205
-1206    if "names" in kwargs:
-1207        names = kwargs.get("names")
-1208    else:
-1209        for f in files:
-1210            if not sep == "":
-1211                se = f.split(".")[0]
-1212                for s in f.split(".")[1:-2]:
-1213                    se += "." + s
-1214                names.append(se.split(sep)[0] + "|r" + se.split(sep)[1])
-1215            else:
-1216                names.append(prefix)
-1217    if 'idl' in kwargs:
-1218        expected_idl = kwargs.get('idl')
-1219    names = sorted(names)
-1220    files = sorted(files)
-1221
-1222    cnfgs = []
-1223    realsamples = []
-1224    imagsamples = []
-1225    repnum = 0
-1226    for file in files:
-1227        with open(path + "/" + file, "rb") as fp:
-1228
-1229            t = fp.read(8)
-1230            kappa = struct.unpack('d', t)[0]
-1231            t = fp.read(8)
-1232            csw = struct.unpack('d', t)[0]
+1197    # test if the input is correct
+1198    if qc not in ['dd', 'ud', 'du', 'uu']:
+1199        raise Exception("Unknown quark conbination!")
+1200
+1201    if corr not in ["gS", "gP", "gA", "gV", "gVt", "lA", "lV", "lVt", "lT", "lTt", "g1", "l1"]:
+1202        raise Exception("Unknown correlator!")
+1203
+1204    if "files" in kwargs:
+1205        known_files = kwargs.get("files")
+1206    else:
+1207        known_files = []
+1208    files = _find_files(path, prefix, "ms5_xsf_" + qc, "dat", known_files=known_files)
+1209
+1210    if "names" in kwargs:
+1211        names = kwargs.get("names")
+1212    else:
+1213        for f in files:
+1214            if not sep == "":
+1215                se = f.split(".")[0]
+1216                for s in f.split(".")[1:-2]:
+1217                    se += "." + s
+1218                names.append(se.split(sep)[0] + "|r" + se.split(sep)[1])
+1219            else:
+1220                names.append(prefix)
+1221    if 'idl' in kwargs:
+1222        expected_idl = kwargs.get('idl')
+1223    names = sorted(names)
+1224    files = sorted(files)
+1225
+1226    cnfgs = []
+1227    realsamples = []
+1228    imagsamples = []
+1229    repnum = 0
+1230    for file in files:
+1231        with open(path + "/" + file, "rb") as fp:
+1232
 1233            t = fp.read(8)
-1234            dF = struct.unpack('d', t)[0]
+1234            kappa = struct.unpack('d', t)[0]
 1235            t = fp.read(8)
-1236            zF = struct.unpack('d', t)[0]
-1237
-1238            t = fp.read(4)
-1239            tmax = struct.unpack('i', t)[0]
-1240            t = fp.read(4)
-1241            bnd = struct.unpack('i', t)[0]
-1242
-1243            placesBI = ["gS", "gP",
-1244                        "gA", "gV",
-1245                        "gVt", "lA",
-1246                        "lV", "lVt",
-1247                        "lT", "lTt"]
-1248            placesBB = ["g1", "l1"]
-1249
-1250            # the chunks have the following structure:
-1251            # confignumber, 10x timedependent complex correlators as doubles, 2x timeindependent complex correlators as doubles
-1252
-1253            chunksize = 4 + (8 * 2 * tmax * 10) + (8 * 2 * 2)
-1254            packstr = '=i' + ('d' * 2 * tmax * 10) + ('d' * 2 * 2)
-1255            cnfgs.append([])
-1256            realsamples.append([])
-1257            imagsamples.append([])
-1258            for t in range(tmax):
-1259                realsamples[repnum].append([])
-1260                imagsamples[repnum].append([])
-1261            if 'idl' in kwargs:
-1262                left_idl = set(expected_idl[repnum])
-1263            while True:
-1264                cnfgt = fp.read(chunksize)
-1265                if not cnfgt:
-1266                    break
-1267                asascii = struct.unpack(packstr, cnfgt)
-1268                cnfg = asascii[0]
-1269                idl_wanted = True
-1270                if 'idl' in kwargs:
-1271                    idl_wanted = (cnfg in expected_idl[repnum])
-1272                    left_idl = left_idl - set([cnfg])
-1273                if idl_wanted:
-1274                    cnfgs[repnum].append(cnfg)
-1275
-1276                    if corr not in placesBB:
-1277                        tmpcorr = asascii[1 + 2 * tmax * placesBI.index(corr):1 + 2 * tmax * placesBI.index(corr) + 2 * tmax]
-1278                    else:
-1279                        tmpcorr = asascii[1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr):1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr) + 2]
-1280
-1281                    corrres = [[], []]
-1282                    for i in range(len(tmpcorr)):
-1283                        corrres[i % 2].append(tmpcorr[i])
-1284                    for t in range(int(len(tmpcorr) / 2)):
-1285                        realsamples[repnum][t].append(corrres[0][t])
-1286                    for t in range(int(len(tmpcorr) / 2)):
-1287                        imagsamples[repnum][t].append(corrres[1][t])
-1288            if 'idl' in kwargs:
-1289                left_idl = list(left_idl)
-1290                if expected_idl[repnum] == left_idl:
-1291                    raise ValueError("None of the idls searched for were found in replikum of file " + file)
-1292                elif len(left_idl) > 0:
-1293                    warnings.warn('Could not find idls ' + str(left_idl) + ' in replikum of file ' + file, UserWarning)
-1294        repnum += 1
-1295    s = "Read correlator " + corr + " from " + str(repnum) + " replika with idls" + str(realsamples[0][t])
-1296    for rep in range(1, repnum):
-1297        s += ", " + str(realsamples[rep][t])
-1298    print(s)
-1299    print("Asserted run parameters:\n T:", tmax, "kappa:", kappa, "csw:", csw, "dF:", dF, "zF:", zF, "bnd:", bnd)
-1300
-1301    # we have the data now... but we need to re format the whole thing and put it into Corr objects.
-1302
-1303    compObs = []
+1236            csw = struct.unpack('d', t)[0]
+1237            t = fp.read(8)
+1238            dF = struct.unpack('d', t)[0]
+1239            t = fp.read(8)
+1240            zF = struct.unpack('d', t)[0]
+1241
+1242            t = fp.read(4)
+1243            tmax = struct.unpack('i', t)[0]
+1244            t = fp.read(4)
+1245            bnd = struct.unpack('i', t)[0]
+1246
+1247            placesBI = ["gS", "gP",
+1248                        "gA", "gV",
+1249                        "gVt", "lA",
+1250                        "lV", "lVt",
+1251                        "lT", "lTt"]
+1252            placesBB = ["g1", "l1"]
+1253
+1254            # the chunks have the following structure:
+1255            # confignumber, 10x timedependent complex correlators as doubles, 2x timeindependent complex correlators as doubles
+1256
+1257            chunksize = 4 + (8 * 2 * tmax * 10) + (8 * 2 * 2)
+1258            packstr = '=i' + ('d' * 2 * tmax * 10) + ('d' * 2 * 2)
+1259            cnfgs.append([])
+1260            realsamples.append([])
+1261            imagsamples.append([])
+1262            for t in range(tmax):
+1263                realsamples[repnum].append([])
+1264                imagsamples[repnum].append([])
+1265            if 'idl' in kwargs:
+1266                left_idl = set(expected_idl[repnum])
+1267            while True:
+1268                cnfgt = fp.read(chunksize)
+1269                if not cnfgt:
+1270                    break
+1271                asascii = struct.unpack(packstr, cnfgt)
+1272                cnfg = asascii[0]
+1273                idl_wanted = True
+1274                if 'idl' in kwargs:
+1275                    idl_wanted = (cnfg in expected_idl[repnum])
+1276                    left_idl = left_idl - set([cnfg])
+1277                if idl_wanted:
+1278                    cnfgs[repnum].append(cnfg)
+1279
+1280                    if corr not in placesBB:
+1281                        tmpcorr = asascii[1 + 2 * tmax * placesBI.index(corr):1 + 2 * tmax * placesBI.index(corr) + 2 * tmax]
+1282                    else:
+1283                        tmpcorr = asascii[1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr):1 + 2 * tmax * len(placesBI) + 2 * placesBB.index(corr) + 2]
+1284
+1285                    corrres = [[], []]
+1286                    for i in range(len(tmpcorr)):
+1287                        corrres[i % 2].append(tmpcorr[i])
+1288                    for t in range(int(len(tmpcorr) / 2)):
+1289                        realsamples[repnum][t].append(corrres[0][t])
+1290                    for t in range(int(len(tmpcorr) / 2)):
+1291                        imagsamples[repnum][t].append(corrres[1][t])
+1292            if 'idl' in kwargs:
+1293                left_idl = list(left_idl)
+1294                if expected_idl[repnum] == left_idl:
+1295                    raise ValueError("None of the idls searched for were found in replikum of file " + file)
+1296                elif len(left_idl) > 0:
+1297                    warnings.warn('Could not find idls ' + str(left_idl) + ' in replikum of file ' + file, UserWarning)
+1298        repnum += 1
+1299    s = "Read correlator " + corr + " from " + str(repnum) + " replika with idls" + str(realsamples[0][t])
+1300    for rep in range(1, repnum):
+1301        s += ", " + str(realsamples[rep][t])
+1302    print(s)
+1303    print("Asserted run parameters:\n T:", tmax, "kappa:", kappa, "csw:", csw, "dF:", dF, "zF:", zF, "bnd:", bnd)
 1304
-1305    for t in range(int(len(tmpcorr) / 2)):
-1306        compObs.append(CObs(Obs([realsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs),
-1307                            Obs([imagsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs)))
+1305    # we have the data now... but we need to re format the whole thing and put it into Corr objects.
+1306
+1307    compObs = []
 1308
-1309    if len(compObs) == 1:
-1310        return compObs[0]
-1311    else:
-1312        return Corr(compObs)
+1309    for t in range(int(len(tmpcorr) / 2)):
+1310        compObs.append(CObs(Obs([realsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs),
+1311                            Obs([imagsamples[rep][t] for rep in range(repnum)], names=names, idl=cnfgs)))
+1312
+1313    if len(compObs) == 1:
+1314        return compObs[0]
+1315    else:
+1316        return Corr(compObs)
 

 
 
diff --git a/docs/search.js b/docs/search.js
index 0d1d976a..957db45d 100644
--- a/docs/search.js
+++ b/docs/search.js
@@ -1,6 +1,6 @@
 window.pdocSearch = (function(){
 /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. 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configuration"),this.buildDefaultConfig(n)}},t.Configuration.prototype.buildDefaultConfig=function(e){this.reset(),e.forEach(function(e){this.config[e]={boost:1,bool:"OR",expand:!1}},this)},t.Configuration.prototype.buildUserConfig=function(e,n){var i="OR",o=!1;if(this.reset(),"bool"in e&&(i=e.bool||i),"expand"in e&&(o=e.expand||o),"fields"in e)for(var r in e.fields)if(n.indexOf(r)>-1){var s=e.fields[r],u=o;void 0!=s.expand&&(u=s.expand),this.config[r]={boost:s.boost||0===s.boost?s.boost:1,bool:s.bool||i,expand:u}}else t.utils.warn("field name in user configuration not found in index instance fields");else this.addAllFields2UserConfig(i,o,n)},t.Configuration.prototype.addAllFields2UserConfig=function(e,t,n){n.forEach(function(n){this.config[n]={boost:1,bool:e,expand:t}},this)},t.Configuration.prototype.get=function(){return this.config},t.Configuration.prototype.reset=function(){this.config={}},lunr.SortedSet=function(){this.length=0,this.elements=[]},lunr.SortedSet.load=function(e){var t=new this;return t.elements=e,t.length=e.length,t},lunr.SortedSet.prototype.add=function(){var e,t;for(e=0;e1;){if(r===e)return o;e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o]}return r===e?o:-1},lunr.SortedSet.prototype.locationFor=function(e){for(var t=0,n=this.elements.length,i=n-t,o=t+Math.floor(i/2),r=this.elements[o];i>1;)e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o];return r>e?o:e>r?o+1:void 0},lunr.SortedSet.prototype.intersect=function(e){for(var t=new lunr.SortedSet,n=0,i=0,o=this.length,r=e.length,s=this.elements,u=e.elements;;){if(n>o-1||i>r-1)break;s[n]!==u[i]?s[n]u[i]&&i++:(t.add(s[n]),n++,i++)}return t},lunr.SortedSet.prototype.clone=function(){var e=new lunr.SortedSet;return e.elements=this.toArray(),e.length=e.elements.length,e},lunr.SortedSet.prototype.union=function(e){var t,n,i;this.length>=e.length?(t=this,n=e):(t=e,n=this),i=t.clone();for(var o=0,r=n.toArray();oWhat is pyerrors?\n\n
pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
\n\n
    \n
  • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
    • \n
  • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
    • \n
  • coherent error propagation for data from different Markov chains.
    • \n
  • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
    • \n
  • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
    • \n
\n\n
More detailed examples can found in the GitHub repository \"badge\".
\n\n
If you use pyerrors for research that leads to a publication please consider citing:
\n\n
    \n
  • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
    • \n
  • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
    • \n
  • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
    • \n
\n\n
and
\n\n
    \n
  • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
    • \n
\n\n
where applicable.
\n\n
There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
\n\n
Installation
\n\n
Install the most recent release using pip and pypi:
\n\n
\n
python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
\n
\n\n
Install the most recent release using conda and conda-forge:
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conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
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Install the current develop version:
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python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
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(Also works for any feature branch).
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Basic example
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import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
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The Obs class
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pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
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import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
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Error propagation
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When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
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The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
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import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
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Error estimation
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The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
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my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.
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We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
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my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
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If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.
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Exponential tails
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Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
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my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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For the full API see pyerrors.obs.Obs.gamma_method.
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Multiple ensembles/replica
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Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
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obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
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Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
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pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
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obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
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Error estimation for multiple ensembles
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In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
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pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
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In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
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Irregular Monte Carlo chains
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Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
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# Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
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Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
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Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
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For the full API see pyerrors.obs.Obs.
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Correlators
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When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
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my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
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In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
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my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
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The individual entries of a correlator can be accessed via slicing
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print(my_corr[3])\n> 0.3227(33)\n
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Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
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my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
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pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
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  • Corr.gamma_method applies the gamma method to all entries of the correlator.
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  • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
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  • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
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  • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
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  • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
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  • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
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  • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
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  • Corr.plateau extracts a plateau value from the correlator in a given range.
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  • Corr.roll periodically shifts the correlator.
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  • Corr.reverse reverses the time ordering of the correlator.
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  • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
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  • Corr.reweight reweights the correlator.
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pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
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For the full API see pyerrors.correlators.Corr.
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Complex valued observables
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pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
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my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
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Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
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my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
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The Covobs class
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In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
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This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
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import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
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The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
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Correlated auxiliary data is defined similarly to above, e.g., via
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RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
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where RAP now is a list of two Obs that contains the two correlated parameters.
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Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
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o.covobs[k].grad\n
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Error propagation in iterative algorithms
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pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
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Least squares fits
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Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
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Fit functions have to be of the following form
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import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
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It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
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Fits can then be performed via
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fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
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where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
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Data stored in Corr objects can be fitted directly using the Corr.fit method.
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my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
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this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
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For fit functions with multiple independent variables the fit function can be of the form
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def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
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pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.
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For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.
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Total least squares fits
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pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in odrpack, see pyerrors.fits.total_least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
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For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
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Matrix operations
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pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
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  • inv for the matrix inverse.
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  • cholseky for the Cholesky decomposition.
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  • det for the matrix determinant.
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  • eigh for eigenvalues and eigenvectors of hermitean matrices.
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  • eig for eigenvalues of general matrices.
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  • pinv for the Moore-Penrose pseudoinverse.
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  • svd for the singular-value-decomposition.
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For the full API see pyerrors.linalg.
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Export data
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The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
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  • What observables are stored? Possibly: How exactly are they defined.
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  • How does each single ensemble or external quantity contribute to the error of the observable?
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  • Who did write the file when and on which machine?
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This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
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my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
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The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
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json.gz format specification
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The first entries of the file provide optional auxiliary information:
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  • program is a string that indicates which program was used to write the file.
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  • version is a string that specifies the version of the format.
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  • who is a string that specifies the user name of the creator of the file.
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  • date is a string and contains the creation date of the file.
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  • host is a string and contains the hostname of the machine where the file has been written.
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  • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
    • \n
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The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
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Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
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  • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
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  • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
    • \n
  • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
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  • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
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  • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
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  • data is an array that contains the data from MC chains. We will define it below.
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  • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
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The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
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  • id, a string that contains the name of the ensemble
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  • replica, an array that contains an entry per replica of the ensemble.
    • \n
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Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
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Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
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The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
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  • id, a string that identifies the covariance matrix
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  • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
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  • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
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  • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
    • \n
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A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
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Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
\n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
\n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
The class for a correlator (time dependent sequence of pe.Obs).
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Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
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The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
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The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.
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Initialization
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A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs
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corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
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A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects
\n\n
\n
matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
\n
\n\n
or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
Initialize a Corr object.
\n\n
Parameters
\n\n
    \n
  • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
    • \n
  • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
    • \n
  • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
    • \n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
We need to project the Correlator with a Vector to get a single value at each timeslice.
\n\n
The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
\n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
\n\n
Parameters
\n\n
    \n
  • i (int):\nFirst index to be picked.
    • \n
  • j (int):\nSecond index to be picked.
    • \n
\n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
Outputs the correlator in a plotable format.
\n\n
Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
Symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
Anti-symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
Checks whether a correlator matrices is symmetric on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "
Calculates the per-timeslice trace of a correlator matrix.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\n
The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
\n\n
\n
C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
\n
\n\n
Parameters
\n\n
    \n
  • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
    • \n
  • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
    • \n
  • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
      \n
    • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
      • \n
    • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
      • \n
    • None: The GEVP is solved only at ts, no sorting is necessary
      • \n
    • \n
  • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
    • \n
\n\n
Other Parameters
\n\n
    \n
  • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
    • \n
  • method (str):\nMethod used to solve the GEVP.\n
      \n
    • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
      • \n
    • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
      • \n
    • \n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
Determines the eigenvalue of the GEVP by solving and projecting the correlator
\n\n
Parameters
\n\n
    \n
  • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
    • \n
  • All other parameters are identical to the ones of Corr.GEVP.
    • \n
\n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
Constructs an NxN Hankel matrix
\n\n
C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
\n\n
Parameters
\n\n
    \n
  • N (int):\nDimension of the Hankel matrix
    • \n
  • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
    • \n
\n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
Periodically shift the correlator by dt timeslices
\n\n
Parameters
\n\n
    \n
  • dt (int):\nnumber of timeslices
    • \n
\n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
Reverse the time ordering of the Corr
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
Thin out a correlator to suppress correlations
\n\n
Parameters
\n\n
    \n
  • spacing (int):\nKeep only every 'spacing'th entry of the correlator
    • \n
  • offset (int):\nOffset the equal spacing
    • \n
\n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
Correlate the correlator with another correlator or Obs
\n\n
Parameters
\n\n
    \n
  • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
    • \n
\n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
Reweight the correlator.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
    • \n
\n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
Return the time symmetry average of the correlator and its partner
\n\n
Parameters
\n\n
    \n
  • partner (Corr):\nTime symmetry partner of the Corr
    • \n
  • parity (int):\nParity quantum number of the correlator, can be +1 or -1
    • \n
\n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
Return the first derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
Return the second derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n    - symmetric (default)\n        $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n    - big_symmetric\n        $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n    - improved\n        $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n    - log\n        $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
Returns the effective mass of the correlator as correlator object
\n\n
Parameters
\n\n
    \n
  • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
    • \n
  • guess (float):\nguess for the root finder, only relevant for the root variant
    • \n
\n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
Fits function to the data
\n\n
Parameters
\n\n
    \n
  • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
    • \n
  • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
    • \n
  • silent (bool):\nDecides whether output is printed to the standard output.
    • \n
\n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
Extract a plateau value from a Corr object
\n\n
Parameters
\n\n
    \n
  • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
    • \n
  • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
    • \n
  • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
    • \n
\n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
Sets the attribute prange of the Corr object.
\n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
Plots the correlator using the tag of the correlator as label if available.
\n\n
Parameters
\n\n
    \n
  • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
    • \n
  • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
    • \n
  • logscale (bool):\nSets y-axis to logscale.
    • \n
  • plateau (Obs):\nPlateau value to be visualized in the figure.
    • \n
  • fit_res (Fit_result):\nFit_result object to be visualized.
    • \n
  • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
    • \n
  • ylabel (str):\nLabel for the y-axis.
    • \n
  • save (str):\npath to file in which the figure should be saved.
    • \n
  • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
    • \n
  • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
    • \n
  • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
    • \n
  • title (string):\nOptional title of the figure.
    • \n
\n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\n
Parameters
\n\n
    \n
  • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
    • \n
\n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
Dumps the Corr into a file of chosen type
\n\n
Parameters
\n\n
    \n
  • filename (str):\nName of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
\n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
Project large correlation matrix to lowest states
\n\n
This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
\n\n
Parameters
\n\n
    \n
  • Ntrunc (int):\nRank of the target matrix.
    • \n
  • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
    • \n
  • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
    • \n
  • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
    • \n
\n\n
Notes
\n\n
We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
\n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
Initialize Covobs object.
\n\n
Parameters
\n\n
    \n
  • mean (float):\nMean value of the new Obs
    • \n
  • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
Return the variance (= square of the error) of the Covobs
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "
\n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
\n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n       [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n       [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "
\n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n       [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "
\n", "default_value": "array([[[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n        [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n        [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n        [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],\n        [ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "
\n", "default_value": "array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "
\n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
Rank-3 epsilon tensor
\n\n
Based on https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
    • \n
\n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
Rank-4 epsilon tensor
\n\n
Extension of https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
    • \n
\n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
Returns gamma matrix in Grid labeling.
\n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
\n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
Represents fit results.
\n\n
Attributes
\n\n
    \n
  • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
    • \n
  • chisquare_by_dof (float):\nreduced chisquare.
    • \n
  • p_value (float):\np-value of the fit
    • \n
  • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
    • \n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "
\n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x).\n    ```
\n\n
Parameters
\n\n
    \n
  • For an uncombined fit:
    • \n
  • x (list):\nlist of floats.
    • \n
  • y (list):\nlist of Obs.
    • \n
  • func (object):\nfit function, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • OR For a combined fit:
    • \n
  • x (dict):\ndict of lists.
    • \n
  • y (dict):\ndict of lists of Obs.
    • \n
  • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n        \"b\": func_b}
    \n\n
    def func_a(a, x):\n    return a[1] * anp.exp(-a[0] * x)
    \n\n
    def func_b(a, x):\n    return a[2] * anp.exp(-a[0] * x)
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
    • \n
  • silent (bool, optional):\nIf True all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
    • \n
  • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg\u2013Marquardt is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
    • \n
  • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg\u2013Marquardt) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
    • \n
  • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
    • \n
  • inv_chol_cov_matrix [array,list], optional: array: shape = (number of y values) X (number of y values)\nlist:   for an uncombined fit: [\"\"]\nfor a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order\nIf correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.\nThe matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be\nused to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct\nordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
    • \n
  • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
    • \n
  • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
    • \n
  • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
  • n_parms (int, optional):\nNumber of fit parameters. Overrides automatic detection of parameter count.\nUseful when autodetection fails. Must match the length of initial_guess or priors (if provided).
    • \n
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n\n
Examples
\n\n
\n
>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set\n>>> import numpy as np\n>>> from scipy.stats import norm\n>>> from scipy.linalg import cholesky\n>>> import pyerrors as pe\n>>> # generating the random data set\n>>> num_samples = 400\n>>> N = 3\n>>> x = np.arange(N)\n>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> r = r1 = r2 = np.zeros((N, N))\n>>> y = {}\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix\n>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix\n>>> c = cholesky(r, lower=True)\n>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined\n>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built\n>>> x_dict = {}\n>>> y_dict = {}\n>>> chol_inv_dict = {}\n>>> data = []\n>>> for key in y.keys():\n>>>    x_dict[key] = x\n>>>    for i in range(N):\n>>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data\n>>>    [o.gamma_method() for o in data]\n>>>    corr = pe.covariance(data, correlation=True)\n>>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))\n>>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below\n>>> y_dict = {'a': data[:3], 'b': data[3:]}\n>>> # common fit parameter p[0] in combined fit\n>>> def fit1(p, x):\n>>>    return p[0] + p[1] * x\n>>> def fit2(p, x):\n>>>    return p[0] + p[2] * x\n>>> fitf_dict = {'a': fit1, 'b':fit2}\n>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])\nFit with 3 parameters\nMethod: Levenberg-Marquardt\n`ftol` termination condition is satisfied.\nchisquare/d.o.f.: 0.5388013574561786 # random\nfit parameters [1.11897846 0.96361162 0.92325319] # random\n
\n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\n
Parameters
\n\n
    \n
  • x (list):\nlist of Obs, or a tuple of lists of Obs
    • \n
  • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
    • \n
  • func (object):\nfunc has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • silent (bool, optional):\nIf True all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
    • \n
  • expected_chisquare (bool):\nIf True prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentiation instead of automatic differentiation to perform the error propagation (default False).
    • \n
  • n_parms (int, optional):\nNumber of fit parameters. Overrides automatic detection of parameter count.\nUseful when autodetection fails. Must match the length of initial_guess (if provided).
    • \n
\n\n
Notes
\n\n
Based on the odrpack orthogonal distance regression library.
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
Performs a linear fit to y = n + m * x and returns two Obs n, m.
\n\n
Parameters
\n\n
    \n
  • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
    • \n
  • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
    • \n
\n\n
Returns
\n\n
    \n
  • fit_parameters (list[Obs]):\nLIist of fitted observables.
    • \n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
Generates a plot which compares the fit to the data and displays the corresponding residuals
\n\n
For uncorrelated data the residuals are expected to be distributed ~N(0,1).
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\n
Returns
\n\n
    \n
  • err (np.array(Obs)):\nError band for an array of sample values x
    • \n
\n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\n
Parameters
\n\n
    \n
  • objects (list):\nList of fit results to include in the analysis (optional).
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
\n\n
Jackknife samples
\n\n
For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
\n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
\n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
Extract generic MCMC data from a bdio file
\n\n
read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • data (List[Obs]):\nExtracted data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
Write Obs to a bdio file according to ADerrors conventions
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • success (int):\nreturns 0 is successful
    • \n
\n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
Extract mesons data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n\n
Returns
\n\n
    \n
  • data (dict):\nExtracted meson data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
Extract dSdm data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, kappa)
\n\n
read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
\n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML formatted string of the input data
    • \n
\n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
Import a list of Obs from a string in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • content (str):\nXML string containing the data
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator |is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML string generated from the data
    • \n
\n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
\n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "
Read hadrons hdf5 file and extract entry based on attributes.
\n\n
Parameters
\n\n
    \n
  • filestem (str):\nFull namestem of the files to read, including the full path.
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • group (str):\nlabel of the group to be extracted.
    • \n
  • attrs (dict or int):\nDictionary containing the attributes. For example
    \n\n
    \n
    attrs = {"gamma_snk": "Gamma5",\n         "gamma_src": "Gamma5"}\n
    \n
    \n\n
    Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
Read hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
    • \n
  • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "
Read hadrons FlowObservables hdf5 file and extract t0
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
\n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
Read hadrons DistillationContraction hdf5 files in given directory structure
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the directories to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict):\nextracted DistillationContration data
    • \n
\n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
ndarray(shape, dtype=float, buffer=None, offset=0, strides=None, order=None)
\n\n
An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
\n\n
Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
\n\n
For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
\n\n
Parameters
\n\n
    \n
  • (for the __new__ method; see Notes below)
    • \n
  • shape (tuple of ints):\nShape of created array.
    • \n
  • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.\nDefault is numpy.float64.
    • \n
  • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
    • \n
  • offset (int, optional):\nOffset of array data in buffer.
    • \n
  • strides (tuple of ints, optional):\nStrides of data in memory.
    • \n
  • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
    • \n
\n\n
Attributes
\n\n
    \n
  • T (ndarray):\nTranspose of the array.
    • \n
  • data (buffer):\nThe array's elements, in memory.
    • \n
  • dtype (dtype object):\nDescribes the format of the elements in the array.
    • \n
  • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
    • \n
  • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
    • \n
  • imag (ndarray):\nImaginary part of the array.
    • \n
  • real (ndarray):\nReal part of the array.
    • \n
  • size (int):\nNumber of elements in the array.
    • \n
  • itemsize (int):\nThe memory use of each array element in bytes.
    • \n
  • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
    • \n
  • ndim (int):\nThe array's number of dimensions.
    • \n
  • shape (tuple of ints):\nShape of the array.
    • \n
  • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
    • \n
  • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
    • \n
  • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
    • \n
\n\n
See Also
\n\n
array:  Construct an array.
\nzeros:  Create an array, each element of which is zero.
\nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
\ndtype:  Create a data-type.
\nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
\n\n
Notes
\n\n
There are two modes of creating an array using __new__:
\n\n
    \n
  1. If buffer is None, then only shape, dtype, and order\nare used.
    2. \n
  2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
    3. \n
\n\n
No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
\n\n
Examples
\n\n
These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
\n\n
First mode, buffer is None:
\n\n
\n
>>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
\n
\n\n
Second mode:
\n\n
\n
>>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
\n
\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
Gamma_5 hermitean conjugate
\n\n
Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
\n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
Read hadrons ExternalLeg hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Npr_matrix):\nread Cobs-matrix
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
Read hadrons Bilinear hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict[Npr_matrix]):\nextracted Bilinears
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • vertices (list):\nVertex functions to be extracted.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict):\nextracted fourquark matrizes
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
\n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
\n\n
Returns
\n\n
    \n
  • json_string (str):\nString for export to .json(.gz) file
    • \n
\n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • Null
    • \n
\n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
Reconstruct a list of Obs or structures containing Obs from a json string.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • json_string (str):\njson string containing the data.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
Import a list of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
Export a dict of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
Import a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
    • \n
\n\n
Returns
\n\n
    \n
  • data (Obs / list / Corr):\nRead data
    • \n
  • or
    • \n
  • data (dict):\nRead data and meta-data
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
\n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "
Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
\n\n
It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
\n\n
A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
\n\n
Parameters
\n\n
    \n
  • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
    • \n
  • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
    • \n
\n\n
Returns
\n\n
    \n
  • root (Obs):\nThe root of the data series.
    • \n
\n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
Read pbp format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of observables read
    • \n
\n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
\n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
Read rwms format from given folder structure. Returns a list of length nrw
\n\n
Parameters
\n\n
    \n
  • path (str):\npath that contains the data files
    • \n
  • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
    • \n
  • version (str):\nversion of openQCD, default 2.0
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • print_err (bool):\nPrint additional information that is useful for debugging.
    • \n
\n\n
Returns
\n\n
    \n
  • rwms (Obs):\nReweighting factors read
    • \n
\n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • postfix (str):\nPostfix of measurement file (Default: ms)
    • \n
  • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • t0 (Obs):\nExtracted t0
    • \n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "
Extract w0/a from given .ms.dat files. Returns w0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt -  (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • postfix (str):\nPostfix of measurement file (Default: ms)
    • \n
  • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • w0 (Obs):\nExtracted w0
    • \n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
Read the topologial charge based on openQCD gradient flow measurements.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nEither openQCD or sfqcd, depending on the data.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
  • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Obs):\nRead topological charge
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\n
Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
Returns the projection to the topological charge sector defined by target.
\n\n
Parameters
\n\n
    \n
  • path (Obs):\nTopological charge.
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
Constructs reweighting factors to a specified topological sector.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
    • \n
  • r_stop (list):\nlast configurations that need to be read (per replicum)
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
\n\n
Parameters
\n\n
    \n
  • path (str):\nThe directory to search for the files in.
    • \n
  • prefix (str):\nThe prefix to match the files against.
    • \n
  • qc (str):\nThe quark combination extension to match the files against.
    • \n
  • corr (str):\nThe correlator to extract data for.
    • \n
  • sep (str, optional):\nThe separator to use when parsing the replika names.
    • \n
  • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
    \n\n
      \n
    • names (List[str]): A list of names to use for the replicas.
      • \n
    • files (List[str]): A list of files to read data from.
      • \n
    • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
      • \n
    • \n
\n\n
Returns
\n\n
    \n
  • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
    • \n
  • or
    • \n
  • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
    • \n
\n\n
Raises
\n\n
    \n
  • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
    • \n
  • IOError: If there is an error reading a file.
    • \n
  • struct.error: If there is an error unpacking binary data.
    • \n
\n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
\n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
Write DataFrame including Obs or Corr valued columns to sqlite database.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be written to the database.
    • \n
  • table_name (str):\nName of the table in the database.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
    • \n
  • gz (bool):\nIf True the json strings are gzipped.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\n
Parameters
\n\n
    \n
  • sql (str):\nSQL query to be executed.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\n
Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be dumped to a file.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
\n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "
\n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
Read sfcf files from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type (str):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf (int):\nID of wave function
    • \n
  • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
    • \n
\n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tcfg_func=None,\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "
Read sfcf files from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf_list (int):\nID of wave function
    • \n
  • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
  • rep_string (str):\nSeparator of ensemble name and replicum. Example: In \"ensAr0\", \"r\" would be the separator string.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n    dict[key] = list[Obs]\n    where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n    dict[name][quarks][offset][wf][wf2] = list[Obs]
    • \n
\n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tcfg_func=None,\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
Utilities for the input
\n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "
Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.
\n\n
Parameters
\n\n
    \n
  • ll (list):\nlist to sort
    • \n
\n\n
Returns
\n\n
    \n
  • ll (list):\nsorted list
    • \n
\n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
Checks if list of configurations is contained in an idl
\n\n
Parameters
\n\n
    \n
  • idl (range or list):\nidl of the current replicum
    • \n
  • che (list):\nlist of configurations to be checked against
    • \n
\n\n
Returns
\n\n
    \n
  • miss_str (str):\nstring with integers of which idls are missing
    • \n
\n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "
Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
\n\n
Parameters
\n\n
    \n
  • path (str):\nmeasurement path, same as for sfcf read method
    • \n
  • param_hash (str):\nexpected parameter hash
    • \n
  • prefix (str):\ndata prefix to find the appropriate replicum folders in path
    • \n
  • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
    • \n
\n\n
Returns
\n\n
    \n
  • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
    • \n
\n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "
\n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "
Performs a (one-dimensional) numeric integration of f(p, x) from a to b.
\n\n
The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
\n\n
Parameters
\n\n
    \n
  • func (object):\nfunction to integrate, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(p, x):\n    return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
    \n
    \n\n
    where x is the integration variable.
    • \n
  • p (list of floats or Obs):\nparameters of the function func.
    • \n
  • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
    • \n
  • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
    • \n
  • All parameters of scipy.integrate.quad
    • \n
\n\n
Returns
\n\n
    \n
  • y (Obs):\nThe integral of func from a to b.
    • \n
  • abserr (float):\nAn estimate of the absolute error in the result.
    • \n
  • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
    • \n
  • message: A convergence message.
    • \n
  • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
    • \n
\n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
Matrix multiply all operands.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • This implementation is faster compared to standard multiplication via the @ operator.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
Matrix multiply both operands making use of the jackknife approximation.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • For large matrices this is considerably faster compared to matmul.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
Wrapper for numpy.einsum
\n\n
Parameters
\n\n
    \n
  • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
    • \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
Inverse of Obs or CObs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
Cholesky decomposition of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
Determinant of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "
Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
\n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "
Print information about version of python, pyerrors and dependencies.
\n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
pyerrors wrapper for the errorbars method of matplotlib
\n\n
Parameters
\n\n
    \n
  • x (list):\nA list of x-values which can be Obs.
    • \n
  • y (list):\nA list of y-values which can be Obs.
    • \n
  • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
    • \n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.12.13/x64/lib/python3.12/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
Dump object into pickle file.
\n\n
Parameters
\n\n
    \n
  • obj (object):\nobject to be saved in the pickle file
    • \n
  • name (str):\nname of the file
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
Load object from pickle file.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the file
    • \n
\n\n
Returns
\n\n
    \n
  • object (Obs):\nLoaded Object
    • \n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
Generate an Obs object with given value, dvalue and name for test purposes
\n\n
Parameters
\n\n
    \n
  • value (float):\ncentral value of the Obs to be generated.
    • \n
  • dvalue (float):\nerror of the Obs to be generated.
    • \n
  • name (str):\nname of the ensemble for which the Obs is to be generated.
    • \n
  • samples (int):\nnumber of samples for the Obs (default 1000).
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nGenerated Observable
    • \n
\n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
Generate observables with given covariance and autocorrelation times.
\n\n
Parameters
\n\n
    \n
  • means (list):\nlist containing the mean value of each observable.
    • \n
  • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
    • \n
  • name (str):\nensemble name for the data to be geneated.
    • \n
  • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
    • \n
  • samples (int):\nnumber of samples to be generated for each observable.
    • \n
\n\n
Returns
\n\n
    \n
  • corr_obs (list[Obs]):\nGenerated observable list
    • \n
\n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
\n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
Matrix pencil method to extract k energy levels from data
\n\n
Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\n
Parameters
\n\n
    \n
  • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
    • \n
  • k (int):\nNumber of states to extract (default 1).
    • \n
  • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
    • \n
\n\n
Returns
\n\n
    \n
  • energy_levels (list[Obs]):\nExtracted energy levels
    • \n
\n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
\n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
Class for a general observable.
\n\n
Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
\n\n
Attributes
\n\n
    \n
  • S_global (float):\nStandard value for S (default 2.0)
    • \n
  • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
    • \n
  • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
    • \n
  • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
  • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
    • \n
  • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
Initialize Obs object.
\n\n
Parameters
\n\n
    \n
  • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
    • \n
  • names (list):\nlist of strings labeling the individual samples
    • \n
  • idl (list, optional):\nlist of ranges or lists on which the samples are defined
    • \n
\n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "
\n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "
\n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "
\n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
Output detailed properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • ens_content (bool):\nprint details about the ensembles and replica if true.
    • \n
\n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
Reweight the obs with given rewighting factors.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
Checks whether the observable is zero within 'sigma' standard errors.
\n\n
Parameters
\n\n
    \n
  • sigma (int):\nNumber of standard errors used for the check.
    • \n
  • Works only properly when the gamma method was run.
    • \n
\n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
Checks whether the observable is zero within a given tolerance.
\n\n
Parameters
\n\n
    \n
  • atol (float):\nAbsolute tolerance (for details see numpy documentation).
    • \n
\n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
Plot integrated autocorrelation time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
Plot normalized autocorrelation function time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
Plot replica distribution for each ensemble with more than one replicum.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
Plot derived Monte Carlo history for each ensemble
\n\n
Parameters
\n\n
    \n
  • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
    • \n
\n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
Dump the Obs to a file 'name' of chosen format.
\n\n
Parameters
\n\n
    \n
  • filename (str):\nname of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • description (str):\nDescription for output file, only relevant for json.gz format.
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
Export jackknife samples from the Obs
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
    • \n
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "
Export bootstrap samples from the Obs
\n\n
Parameters
\n\n
    \n
  • samples (int):\nNumber of bootstrap samples to generate.
    • \n
  • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
    • \n
  • save_rng (str):\nSave the random numbers to a file if a path is specified.
    • \n
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
    • \n
\n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
Class for a complex valued observable.
\n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
\n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "
Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\n
See docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "
Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\n
See docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\n
Parameters
\n\n
    \n
  • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
    • \n
  • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
    • \n
  • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
    • \n
\n\n
Notes
\n\n
For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\n
new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
\n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
Reweight a list of observables.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
Correlate two observables.
\n\n
Parameters
\n\n
    \n
  • obs_a (Obs):\nFirst observable
    • \n
  • obs_b (Obs):\nSecond observable
    • \n
\n\n
Notes
\n\n
Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nOnly works if a single ensemble is present in the Obs.\nCurrently only works if ensemble content is identical (this is not strictly necessary).
\n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
Calculates the error covariance matrix of a set of observables.
\n\n
WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
\n\n
The gamma method has to be applied first to all observables.
\n\n
Parameters
\n\n
    \n
  • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
    • \n
  • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
    • \n
  • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
    • \n
  • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
    • \n
\n\n
Notes
\n\n
The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
\n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.invert_corr_cov_cholesky": {"fullname": "pyerrors.obs.invert_corr_cov_cholesky", "modulename": "pyerrors.obs", "qualname": "invert_corr_cov_cholesky", "kind": "function", "doc": "
Constructs a lower triangular matrix chol via the Cholesky decomposition of the correlation matrix corr\n   and then returns the inverse covariance matrix chol_inv as a lower triangular matrix by solving chol * x = inverrdiag.
\n\n
Parameters
\n\n
    \n
  • corr (np.ndarray):\ncorrelation matrix
    • \n
  • inverrdiag (np.ndarray):\ndiagonal matrix, the entries are the inverse errors of the data points considered
    • \n
\n", "signature": "(corr, inverrdiag):", "funcdef": "def"}, "pyerrors.obs.sort_corr": {"fullname": "pyerrors.obs.sort_corr", "modulename": "pyerrors.obs", "qualname": "sort_corr", "kind": "function", "doc": "
Reorders a correlation matrix to match the alphabetical order of its underlying y data.
\n\n
The ordering of the input correlation matrix corr is given by the list of keys kl.\nThe input dictionary yd (with the same keys kl) must contain the corresponding y data\nthat the correlation matrix is based on.\nThis function sorts the list of keys kl alphabetically and sorts the matrix corr\naccording to this alphabetical order such that the sorted matrix corr_sorted corresponds\nto the y data yd when arranged in an alphabetical order by its keys.
\n\n
Parameters
\n\n
    \n
  • corr (np.ndarray):\nA square correlation matrix constructed using the order of the y data specified by kl.\nThe dimensions of corr should match the total number of y data points in yd combined.
    • \n
  • kl (list of str):\nA list of keys that denotes the order in which the y data from yd was used to build the\ninput correlation matrix corr.
    • \n
  • yd (dict of list):\nA dictionary where each key corresponds to a unique identifier, and its value is a list of\ny data points. The total number of y data points across all keys must match the dimensions\nof corr. The lists in the dictionary can be lists of Obs.
    • \n
\n\n
Returns
\n\n
    \n
  • np.ndarray: A new, sorted correlation matrix that corresponds to the y data from yd when arranged alphabetically by its keys.
    • \n
\n\n
Example
\n\n
\n
>>> import numpy as np\n>>> import pyerrors as pe\n>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])\n>>> kl = ['b', 'a']\n>>> yd = {'a': [1, 2], 'b': [3]}\n>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)\n>>> print(sorted_corr)\narray([[1. , 0.3, 0.4],\n       [0.3, 1. , 0.2],\n       [0.4, 0.2, 1. ]])\n
\n
\n", "signature": "(corr, kl, yd):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
Imports jackknife samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
\n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "
Imports bootstrap samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
  • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
    • \n
\n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
Combine all observables in list_of_obs into one new observable.\nThis allows to merge Obs that have been computed on multiple replica\nof the same ensemble.\nIf you like to merge Obs that are based on several ensembles, please\naverage them yourself.
\n\n
Parameters
\n\n
    \n
  • list_of_obs (list):\nlist of the Obs object to be combined
    • \n
\n\n
Notes
\n\n
It is not possible to combine obs which are based on the same replicum
\n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
Create an Obs based on mean(s) and a covariance matrix
\n\n
Parameters
\n\n
    \n
  • mean (list of floats or float):\nN mean value(s) of the new Obs
    • \n
  • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
\n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
Finds the root of the function func(x, d) where d is an Obs.
\n\n
Parameters
\n\n
    \n
  • d (Obs):\nObs passed to the function.
    • \n
  • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
    \n\n
    \n
    import autograd.numpy as anp\ndef root_func(x, d):\n    return anp.exp(-x ** 2) - d\n
    \n
    • \n
  • guess (float):\nInitial guess for the minimization.
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nObs valued root of the function.
    • \n
\n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "
\n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "
beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
beta(a, b, out=None)
\n\n
Beta function.
\n\n
This function is defined in 1 as
\n\n
$$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n        = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
\n\n
where \\( \\Gamma \\) is the gamma function.
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nReal-valued arguments
    • \n
  • out (ndarray, optional):\nOptional output array for the function result
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the beta function
    • \n
\n\n
See Also
\n\n
gamma:  the gamma function
\nbetainc:   the regularized incomplete beta function
\nbetaln:  the natural logarithm of the absolute\nvalue of the beta function  
\n\n
References
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
The beta function relates to the gamma function by the\ndefinition given above:
\n\n
\n
>>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
\n
\n\n
As this relationship demonstrates, the beta function\nis symmetric:
\n\n
\n
>>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
\n
\n\n
This function satisfies \\( B(1, b) = 1/b \\):
\n\n
\n
>>> sc.beta(1, 4)\n0.25\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
betainc(a, b, x, out=None)
\n\n
Regularized incomplete beta function.
\n\n
Computes the regularized incomplete beta function, defined as 1:
\n\n
$$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
\n\n
for \\( 0 \\leq x \\leq 1 \\).
\n\n
This function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nPositive, real-valued parameters
    • \n
  • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the regularized incomplete beta function
    • \n
\n\n
See Also
\n\n
beta()`\nbeta`, `function`  \nbetaincinv()\ninverse, of, the, regularized, incomplete, beta, function
\nbetaincc()`\ncomplement`, `of`, `the`, `regularized`, `incomplete`, `beta`, `function`  \nscipy.stats.beta()\nbeta, distribution  
\n\n
Notes
\n\n
The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula.  When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms.  One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).
\n\n
betainc(a, b, x) is treated as a two parameter family of functions\nof a single variable x, rather than as a function of three variables.\nThis impacts only the limiting cases a = 0, b = 0, a = inf,\nb = inf.
\n\n
In general
\n\n
$$\\lim_{(a, b) \\rightarrow (a_0, b_0)} \\mathrm{betainc}(a, b, x)$$
\n\n
is treated as a pointwise limit in x. Thus for example,\nbetainc(0, b, 0) equals 0 for b > 0, although it would be\nindeterminate when considering the simultaneous limit (a, x) -> (0+, 0+).
\n\n
This function wraps the ibeta routine from the\nBoost Math C++ library 2.
\n\n
Array API Standard Support
\n\n
betainc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Let \\( B(a, b) \\) be the beta function.
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).
\n\n
\n
>>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
\n
\n\n
It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:
\n\n
\n
>>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
\n
\n\n
This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
\n\n
\n
>>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
    \n
    2. \n\n
  2. \n
    The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/. ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
betaln(a, b, out=None)
\n\n
Natural logarithm of absolute value of beta function.
\n\n
Computes ln(abs(beta(a, b))).
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nPositive, real-valued parameters
    • \n
  • out (ndarray, optional):\nOptional output array for function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the betaln function
    • \n
\n\n
See Also
\n\n
gamma()`\nthe`, `gamma`, `function`  \nbetainc()\nthe, regularized, incomplete, beta, function
\n`beta()\nthe,beta,function`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
betaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import betaln, beta\n
\n
\n\n
Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):
\n\n
\n
>>> betaln(3, 4)\n-4.0943445622221\n
\n
\n\n
\n
>>> np.log(beta(3, 4))\n-4.0943445622221\n
\n
\n\n
In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.
\n\n
\n
>>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
\n
\n\n
We can compute the logarithm of beta(a, b) by using betaln:
\n\n
\n
>>> betaln(a, b)\n-804.3069951764146\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "
Polygamma functions.
\n\n
Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
    • \n
  • x (array_like):\nReal valued input
    • \n
\n\n
Returns
\n\n
    \n
  • ndarray: Function results
    • \n
\n\n
See Also
\n\n
`digamma()\n..`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
polygamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST, Digital Library of Mathematical Functions,\n    https://dlmf.nist.gov/5.15
\n\n
Examples
\n\n
\n
>>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "
psi(z, out=None)
\n\n
The digamma function.
\n\n
The logarithmic derivative of the gamma function evaluated at z.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex argument.
    • \n
  • out (ndarray, optional):\nArray for the computed values of psi.
    • \n
\n\n
Returns
\n\n
    \n
  • digamma (scalar or ndarray):\nComputed values of psi.
    • \n
\n\n
Notes
\n\n
For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 1 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 1 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 2, so the\nfunction should maintain full accuracy around the origin.
\n\n
Array API Standard Support
\n\n
psi has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n
\n
>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
    \n
    2. \n\n
  2. \n
    Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
psi(z, out=None)
\n\n
The digamma function.
\n\n
The logarithmic derivative of the gamma function evaluated at z.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex argument.
    • \n
  • out (ndarray, optional):\nArray for the computed values of psi.
    • \n
\n\n
Returns
\n\n
    \n
  • digamma (scalar or ndarray):\nComputed values of psi.
    • \n
\n\n
Notes
\n\n
For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 1 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 1 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 2, so the\nfunction should maintain full accuracy around the origin.
\n\n
Array API Standard Support
\n\n
psi has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n
\n
>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
    \n
    2. \n\n
  2. \n
    Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
gamma(z, out=None)
\n\n
gamma function.
\n\n
The gamma function is defined as
\n\n
$$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
\n\n
for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex valued argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the gamma function
    • \n
\n\n
Notes
\n\n
The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
\n\n
The gamma function has poles at non-negative integers and the sign\nof infinity as z approaches each pole depends upon the direction in\nwhich the pole is approached. For this reason, the consistent thing\nis for gamma(z) to return NaN at negative integers, and to return\n-inf when x = -0.0 and +inf when x = 0.0, using the signbit of zero\nto signify the direction in which the origin is being approached. This\nis for instance what is recommended for the gamma function in annex F\nentry 9.5.4 of the Iso C 99 standard [isoc99]_.
\n\n
Prior to SciPy version 1.15, scipy.special.gamma(z) returned +inf\nat each pole. This was fixed in version 1.15, but with the following\nconsequence. Expressions where gamma appears in the denominator\nsuch as
\n\n
gamma(u) * gamma(v) / (gamma(w) * gamma(x))
\n\n
no longer evaluate to 0 if the numerator is well defined but there is a\npole in the denominator. Instead such expressions evaluate to NaN. We\nrecommend instead using the function rgamma for the reciprocal gamma\nfunction in such cases. The above expression could for instance be written\nas
\n\n
gamma(u) * gamma(v) * (rgamma(w) * rgamma(x))
\n\n
Array API Standard Support
\n\n
gamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1\n.. [isoc99] https://www.open-std.org/jtc1/sc22/wg14/www/docs/n1256.pdf
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
\n
\n\n
\n
>>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
\n
\n\n
\n
>>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
\n
\n\n
\n
>>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
\n
\n\n
Plot gamma(x) for real x
\n\n
\n
>>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
\n
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "
gammaln(x, out=None)
\n\n
Logarithm of the absolute value of the gamma function.
\n\n
Defined as
\n\n
$$\\ln(\\lvert\\Gamma(x)\\rvert)$$
\n\n
where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the log of the absolute value of gamma
    • \n
\n\n
See Also
\n\n
gammasgn()`\nsign`, `of`, `the`, `gamma`, `function`  \nloggamma()\nprincipal, branch, of, the, logarithm, of, the, gamma, function  
\n\n
Notes
\n\n
It is the same function as the Python standard library function\nmath.lgamma().
\n\n
When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).
\n\n
For complex-valued log-gamma, use loggamma instead of gammaln.
\n\n
Array API Standard Support
\n\n
gammaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import scipy.special as sc\n
\n
\n\n
It has two positive zeros.
\n\n
\n
>>> sc.gammaln([1, 2])\narray([0., 0.])\n
\n
\n\n
It has poles at nonpositive integers.
\n\n
\n
>>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
\n
\n\n
It asymptotically approaches x * log(x) (Stirling's formula).
\n\n
\n
>>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "
gammainc(a, x, out=None)
\n\n
Regularized lower incomplete gamma function.
\n\n
It is defined as
\n\n
$$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
\n\n
for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nPositive parameter
    • \n
  • x (array_like):\nNonnegative argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the lower incomplete gamma function
    • \n
\n\n
See Also
\n\n
gammaincc()`\nregularized`, `upper`, `incomplete`, `gamma`, `function`  \ngammaincinv()\ninverse, of, the, regularized, lower, incomplete, gamma, function
\n`gammainccinv()\ninverse,of,the,regularized,upper,incomplete,gamma,function`  
\n\n
Notes
\n\n
The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.
\n\n
The implementation largely follows that of [boost]_.
\n\n
Array API Standard Support
\n\n
gammainc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
\n\n
\n
>>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
\n
\n\n
It is equal to one minus the upper incomplete gamma function.
\n\n
\n
>>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "
gammaincc(a, x, out=None)
\n\n
Regularized upper incomplete gamma function.
\n\n
It is defined as
\n\n
$$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
\n\n
for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nPositive parameter
    • \n
  • x (array_like):\nNonnegative argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the upper incomplete gamma function
    • \n
\n\n
See Also
\n\n
gammainc()`\nregularized`, `lower`, `incomplete`, `gamma`, `function`  \ngammaincinv()\ninverse, of, the, regularized, lower, incomplete, gamma, function
\n`gammainccinv()\ninverse,of,the,regularized,upper,incomplete,gamma,function`  
\n\n
Notes
\n\n
The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.
\n\n
The implementation largely follows that of [boost]_.
\n\n
Array API Standard Support
\n\n
gammaincc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
\n\n
\n
>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
\n
\n\n
It is equal to one minus the lower incomplete gamma function.
\n\n
\n
>>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "
gammasgn(x, out=None)
\n\n
Sign of the gamma function.
\n\n
It is defined as
\n\n
$$\\text{gammasgn}(x) =\n\\begin{cases}\n  +1 & \\Gamma(x) > 0 \\\n  -1 & \\Gamma(x) < 0\n\\end{cases}$$
\n\n
where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Sign of the gamma function
    • \n
\n\n
See Also
\n\n
gamma()`\nthe`, `gamma`, `function`  \ngammaln()\nlog, of, the, absolute, value, of, the, gamma, function
\n`loggamma()\nanalytic,continuation,of,the,log,of,the,gamma,function`  
\n\n
Notes
\n\n
The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).
\n\n
Array API Standard Support
\n\n
gammasgn has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import scipy.special as sc\n
\n
\n\n
It is 1 for x > 0.
\n\n
\n
>>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
\n
\n\n
It alternates between -1 and 1 for negative integers.
\n\n
\n
>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
\n
\n\n
It can be used to compute the gamma function.
\n\n
\n
>>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "
rgamma(z, out=None)
\n\n
Reciprocal of the gamma function.
\n\n
Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex valued input
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Function results
    • \n
\n\n
See Also
\n\n
gamma()`,`,gammaln(),, `loggamma()\n..`  
\n\n
Notes
\n\n
The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.
\n\n
Array API Standard Support
\n\n
rgamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] Nist, Digital Library of Mathematical functions,\n    https://dlmf.nist.gov/5.2#i
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the reciprocal of the gamma function.
\n\n
\n
>>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
\n
\n\n
It is zero at nonpositive integers.
\n\n
\n
>>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
\n
\n\n
It rapidly underflows to zero along the positive real axis.
\n\n
\n
>>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "
Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
\n\n
Parameters
\n\n
    \n
  • a (ndarray):\nThe multivariate gamma is computed for each item of a.
    • \n
  • d (int):\nThe dimension of the space of integration.
    • \n
\n\n
Returns
\n\n
    \n
  • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
    • \n
\n\n
Notes
\n\n
The formal definition of the multivariate gamma of dimension d for a real\na is
\n\n
$$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
\n\n
with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d.  Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).
\n\n
This can be proven to be equal to the much friendlier equation
\n\n
$$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
\n\n
Array API Standard Support
\n\n
multigammaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
\n
\n\n
Verify that the result agrees with the logarithm of the equation\nshown above:
\n\n
\n
>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "
Modified Bessel function of the second kind of integer order n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "
j0(x, out=None)
\n\n
Bessel function of the first kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
    • \n
\n\n
See Also
\n\n
jv()`\nBessel`, `function`, `of`, `real`, `order`, `and`, `complex`, `argument.`  \nspherical_jn()\nspherical, Bessel, functions.  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
\n\n
$$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
\n\n
where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
\n\n
In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\n
This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
\n\n
Array API Standard Support
\n\n
j0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
\n
\n\n
Plot the function from -20 to 20.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
y0(x, out=None)
\n\n
Bessel function of the second kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
    • \n
\n\n
See Also
\n\n
j0()`\nBessel`, `function`, `of`, `the`, `first`, `kind`, `of`, `order`, `0`  \nyv()\nBessel, function, of, the, first, kind  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
\n\n
$$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
\n\n
where \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\n
In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\n
This function is a wrapper for the Cephes 1 routine y0.
\n\n
Array API Standard Support
\n\n
y0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
\n
\n\n
Plot the function from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
j1(x, out=None)
\n\n
Bessel function of the first kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
    • \n
\n\n
See Also
\n\n
jv()`\nBessel`, `function`, `of`, `the`, `first`, `kind`  \nspherical_jn()\nspherical, Bessel, functions.  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
\n\n
This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
\n\n
Array API Standard Support
\n\n
j1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
\n
\n\n
Plot the function from -20 to 20.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
y1(x, out=None)
\n\n
Bessel function of the second kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
    • \n
\n\n
See Also
\n\n
j1()`\nBessel`, `function`, `of`, `the`, `first`, `kind`, `of`, `order`, `1`  \nyn()\nBessel, function, of, the, second, kind
\n`yv()\nBessel,function,of,the,second,kind`  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
\n\n
This function is a wrapper for the Cephes 1 routine y1.
\n\n
Array API Standard Support
\n\n
y1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
\n
\n\n
Plot the function from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
jv(v, z, out=None)
\n\n
Bessel function of the first kind of real order and complex argument.
\n\n
Parameters
\n\n
    \n
  • v (array_like):\nOrder (float).
    • \n
  • z (array_like):\nArgument (float or complex).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
    • \n
\n\n
See Also
\n\n
jve:  \\( J_v \\) with leading exponential behavior stripped off.
\nspherical_jn:  spherical Bessel functions.
\nj0:  faster version of this function for order 0.
\nj1:  faster version of this function for order 1.  
\n\n
Notes
\n\n
For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),
\n\n
$$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
\n\n
J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
\n\n
For negative v values the formula,
\n\n
$$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
\n\n
is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy.  Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).
\n\n
Not to be confused with the spherical Bessel functions (see spherical_jn).
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -10 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
yn(n, x, out=None)
\n\n
Bessel function of the second kind of integer order and real argument.
\n\n
Parameters
\n\n
    \n
  • n (array_like):\nOrder (integer).
    • \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
    • \n
\n\n
See Also
\n\n
yv()`\nFor`, `real`, `order`, `and`, `real`, `or`, `complex`, `argument.`  \ny0()\nfaster, implementation, of, this, function, for, order, 0
\n`y1()\nfaster,implementation,of,this,function,for,order,1`  
\n\n
Notes
\n\n
Wrapper for the Cephes 1 routine yn.
\n\n
The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.
\n\n
Array API Standard Support
\n\n
yn has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
\n
\n\n
Plot the functions of order 0 to 3 from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
i0(x, out=None)
\n\n
Modified Bessel function of order 0.
\n\n
Defined as,
\n\n
$$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
\n\n
where \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float)
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
    • \n
\n\n
See Also
\n\n
iv()`\nModified`, `Bessel`, `function`, `of`, `any`, `order`  \ni0e()\nExponentially, scaled, modified, Bessel, function, of, order, 0  
\n\n
Notes
\n\n
The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\n
This function is a wrapper for the Cephes 1 routine i0.
\n\n
Array API Standard Support
\n\n
i0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
\n
\n\n
Plot the function from -10 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
i1(x, out=None)
\n\n
Modified Bessel function of order 1.
\n\n
Defined as,
\n\n
$$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n       = -\\imath J_1(\\imath x),$$
\n\n
where \\( J_1 \\) is the Bessel function of the first kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float)
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
    • \n
\n\n
See Also
\n\n
iv()`\nModified`, `Bessel`, `function`, `of`, `the`, `first`, `kind`  \ni1e()\nExponentially, scaled, modified, Bessel, function, of, order, 1  
\n\n
Notes
\n\n
The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\n
This function is a wrapper for the Cephes 1 routine i1.
\n\n
Array API Standard Support
\n\n
i1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
\n
\n\n
Plot the function between -10 and 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
iv(v, z, out=None)
\n\n
Modified Bessel function of the first kind of real order.
\n\n
Parameters
\n\n
    \n
  • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
    • \n
  • z (array_like of float or complex):\nArgument.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the modified Bessel function.
    • \n
\n\n
See Also
\n\n
ive:  This function with leading exponential behavior stripped off.
\ni0:  Faster version of this function for order 0.
\ni1:  Faster version of this function for order 1.  
\n\n
Notes
\n\n
For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1.  For larger orders, uniform asymptotic\nexpansions are applied.
\n\n
For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.
\n\n
The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n
$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n
(valid when the real part of z is positive).  For negative v, the\nformula
\n\n
$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\n
is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩
    \n
    2. \n\n
  2. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
ive(v, z, out=None)
\n\n
Exponentially scaled modified Bessel function of the first kind.
\n\n
Defined as::
\n\n
ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
\n\n
For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.
\n\n
Parameters
\n\n
    \n
  • v (array_like of float):\nOrder.
    • \n
  • z (array_like of float or complex):\nArgument.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
    • \n
\n\n
See Also
\n\n
iv:  Modified Bessel function of the first kind
\ni0e:  Faster implementation of this function for order 0
\ni1e:  Faster implementation of this function for order 1  
\n\n
Notes
\n\n
For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.
\n\n
The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n
$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n
(valid when the real part of z is positive).  For negative v, the\nformula
\n\n
$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\n
is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
\n\n
ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.
\n\n
References
\n\n
Examples
\n\n
In the following example iv returns infinity whereas ive still returns\na finite number.
\n\n
\n
>>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
\n
\n\n
Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
\n
\n\n
Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:
\n\n
\n
>>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n
\n
>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
erf(z, out=None)
\n\n
Returns the error function of complex argument.
\n\n
It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nInput array.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • res (scalar or ndarray):\nThe values of the error function at the given points x.
    • \n
\n\n
See Also
\n\n
erfc()`,`,erfinv(),, erfcinv()`,`,wofz(),, erfcx()`,`,erfi()\n..  
\n\n
Notes
\n\n
The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].
\n\n
Array API Standard Support
\n\n
erf has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
erfc(x, out=None)
\n\n
Complementary error function, 1 - erf(x).
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal or complex valued argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the complementary error function
    • \n
\n\n
See Also
\n\n
erf()`,`,erfi(),, erfcx()`,`,dawsn(),, `wofz()\n..`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
erfc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
erfinv(y, out=None)
\n\n
Inverse of the error function.
\n\n
Computes the inverse of the error function.
\n\n
In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
\n\n
Parameters
\n\n
    \n
  • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
    • \n
\n\n
See Also
\n\n
erf()`\nError`, `function`, `of`, `a`, `complex`, `argument`  \nerfc()\nComplementary, error, function,, 1`, `-`, `erf(x)
\n`erfcinv()\nInverse,of,the,complementary,error,function`  
\n\n
Notes
\n\n
This function wraps the erf_inv routine from the\nBoost Math C++ library 1.
\n\n
Array API Standard Support
\n\n
erfinv has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
\n
\n\n
\n
>>> erfinv(0.5)\n0.4769362762044699\n
\n
\n\n
\n
>>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
\n
\n\n
Verify that erf(erfinv(y)) is y.
\n\n
\n
>>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
\n
\n\n
Plot the function:
\n\n
\n
>>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/. ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "
erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
erfcinv(y, out=None)
\n\n
Inverse of the complementary error function.
\n\n
Computes the inverse of the complementary error function.
\n\n
In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
\n\n
It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
\n\n
Parameters
\n\n
    \n
  • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
    • \n
\n\n
See Also
\n\n
erf:  Error function of a complex argument
\nerfc:  Complementary error function, 1 - erf(x)
\nerfinv:  Inverse of the error function  
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
\n
\n\n
\n
>>> erfcinv(0.5)\n0.4769362762044699\n
\n
\n\n
\n
>>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
\n
\n\n
Plot the function:
\n\n
\n
>>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "
logit(x, out=None)
\n\n
Logit ufunc for ndarrays.
\n\n
The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nThe ndarray to apply logit to element-wise.
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
    • \n
\n\n
See Also
\n\n
`expit()\n..`  
\n\n
Notes
\n\n
As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
\n\n
New in version 0.10.0.
\n\n
Array API Standard Support
\n\n
logit has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import logit, expit\n
\n
\n\n
\n
>>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
\n
\n\n
expit is the inverse of logit:
\n\n
\n
>>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
\n
\n\n
Plot logit(x) for x in [0, 1]:
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "
expit(x, out=None)
\n\n
Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.
\n\n
The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)).  It is the inverse of the\nlogit function.
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nThe ndarray to apply expit to element-wise.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
    • \n
\n\n
See Also
\n\n
`logit()\n..`  
\n\n
Notes
\n\n
As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
\n\n
New in version 0.10.0.
\n\n
Array API Standard Support
\n\n
expit has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import expit, logit\n
\n
\n\n
\n
>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
\n
\n\n
logit is the inverse of expit:
\n\n
\n
>>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
\n
\n\n
Plot expit(x) for x in [-6, 6]:
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "
Compute the log of the sum of exponentials of input elements.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nInput array.
    • \n
  • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.
    \n\n
    New in version 0.11.0.
    • \n
  • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.
    \n\n
    New in version 0.12.0.
    • \n
  • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
    \n\n
    New in version 0.15.0.
    • \n
  • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
    \n\n
    New in version 0.16.0.
    • \n
\n\n
Returns
\n\n
    \n
  • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned. If return_sign is True, res contains the log of\nthe absolute value of the argument.
    • \n
  • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm in res.\nIf return_sign is False, only one result is returned.
    • \n
\n\n
See Also
\n\n
numpy.logaddexp`\n..`  \nnumpy.logaddexp2\n..  
\n\n
Notes
\n\n
NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.
\n\n
The logarithm is a multivalued function: for each \\( x \\) there is an\ninfinite number of \\( z \\) such that \\( exp(z) = x \\). The convention\nis to return the \\( z \\) whose imaginary part lies in \\( (-pi, pi] \\).
\n\n
Array API Standard Support
\n\n
logsumexp has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
\n
\n\n
With weights
\n\n
\n
>>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
\n
\n\n
Returning a sign flag
\n\n
\n
>>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
\n
\n\n
Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:
\n\n
\n
>>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
\n
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What is pyerrors?
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pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
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  • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
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  • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
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  • coherent error propagation for data from different Markov chains.
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  • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
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  • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
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More detailed examples can found in the GitHub repository \"badge\".
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If you use pyerrors for research that leads to a publication please consider citing:
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  • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
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  • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
    • \n
  • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
    • \n
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and
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  • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
    • \n
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where applicable.
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There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
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Installation
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Install the most recent release using pip and pypi:
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python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
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Install the most recent release using conda and conda-forge:
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conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
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Install the current develop version:
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python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
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(Also works for any feature branch).
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Basic example
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import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
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The Obs class
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pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
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import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
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Error propagation
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When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
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The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
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import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
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Error estimation
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The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
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my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.
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We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
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my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
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If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.
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Exponential tails
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Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
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my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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For the full API see pyerrors.obs.Obs.gamma_method.
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Multiple ensembles/replica
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Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
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obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
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Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
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pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
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obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
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Error estimation for multiple ensembles
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In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
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pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
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In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
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Irregular Monte Carlo chains
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Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
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# Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
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Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
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Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
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For the full API see pyerrors.obs.Obs.
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Correlators
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When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
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my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
\n
\n\n
In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
\n\n
\n
my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
\n
\n\n
The individual entries of a correlator can be accessed via slicing
\n\n
\n
print(my_corr[3])\n> 0.3227(33)\n
\n
\n\n
Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
\n\n
\n
my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
\n
\n\n
pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
\n\n
    \n
  • Corr.gamma_method applies the gamma method to all entries of the correlator.
    • \n
  • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
    • \n
  • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
    • \n
  • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
    • \n
  • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
    • \n
  • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
    • \n
  • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
    • \n
  • Corr.plateau extracts a plateau value from the correlator in a given range.
    • \n
  • Corr.roll periodically shifts the correlator.
    • \n
  • Corr.reverse reverses the time ordering of the correlator.
    • \n
  • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
    • \n
  • Corr.reweight reweights the correlator.
    • \n
\n\n
pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
\n\n
For the full API see pyerrors.correlators.Corr.
\n\n
Complex valued observables
\n\n
pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
\n\n
\n
my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
\n
\n\n
Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
\n\n
\n
my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
\n
\n\n
The Covobs class
\n\n
In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
\n\n
This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
\n\n
\n
import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
\n
\n\n
The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
\n\n
Correlated auxiliary data is defined similarly to above, e.g., via
\n\n
\n
RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
\n
\n\n
where RAP now is a list of two Obs that contains the two correlated parameters.
\n\n
Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
\n\n
\n
o.covobs[k].grad\n
\n
\n\n
Error propagation in iterative algorithms
\n\n
pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
\n\n
Least squares fits
\n\n
Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
\n\n
Fit functions have to be of the following form
\n\n
\n
import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
\n
\n\n
It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
\n\n
Fits can then be performed via
\n\n
\n
fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
\n
\n\n
where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
\n\n
Data stored in Corr objects can be fitted directly using the Corr.fit method.
\n\n
\n
my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
\n
\n\n
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
\n\n
For fit functions with multiple independent variables the fit function can be of the form
\n\n
\n
def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
\n
\n\n
pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.
\n\n
For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.
\n\n
Total least squares fits
\n\n
pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in odrpack, see pyerrors.fits.total_least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
\n\n
For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
\n\n
Matrix operations
\n\n
pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
\n\n
    \n
  • inv for the matrix inverse.
    • \n
  • cholseky for the Cholesky decomposition.
    • \n
  • det for the matrix determinant.
    • \n
  • eigh for eigenvalues and eigenvectors of hermitean matrices.
    • \n
  • eig for eigenvalues of general matrices.
    • \n
  • pinv for the Moore-Penrose pseudoinverse.
    • \n
  • svd for the singular-value-decomposition.
    • \n
\n\n
For the full API see pyerrors.linalg.
\n\n
Export data
\n\n
\n\n
The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
\n\n
    \n
  • What observables are stored? Possibly: How exactly are they defined.
    • \n
  • How does each single ensemble or external quantity contribute to the error of the observable?
    • \n
  • Who did write the file when and on which machine?
    • \n
\n\n
This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
\n\n
\n
my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
\n
\n\n
The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
\n\n
json.gz format specification
\n\n
The first entries of the file provide optional auxiliary information:
\n\n
    \n
  • program is a string that indicates which program was used to write the file.
    • \n
  • version is a string that specifies the version of the format.
    • \n
  • who is a string that specifies the user name of the creator of the file.
    • \n
  • date is a string and contains the creation date of the file.
    • \n
  • host is a string and contains the hostname of the machine where the file has been written.
    • \n
  • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
    • \n
\n\n
The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
\n\n
Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
\n\n
    \n
  • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
    • \n
  • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
    • \n
  • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
    • \n
  • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
    • \n
  • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
    • \n
  • data is an array that contains the data from MC chains. We will define it below.
    • \n
  • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
    • \n
\n\n
The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
\n\n
    \n
  • id, a string that contains the name of the ensemble
    • \n
  • replica, an array that contains an entry per replica of the ensemble.
    • \n
\n\n
Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
\n\n
Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
\n\n
The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
\n\n
    \n
  • id, a string that identifies the covariance matrix
    • \n
  • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
    • \n
  • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
    • \n
  • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
    • \n
\n\n
A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
\n\n
Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
\n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
\n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
The class for a correlator (time dependent sequence of pe.Obs).
\n\n
Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
\n\n
The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
\n\n
The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.
\n\n
Initialization
\n\n
A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs
\n\n
\n
corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
\n
\n\n
A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects
\n\n
\n
matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
\n
\n\n
or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
Initialize a Corr object.
\n\n
Parameters
\n\n
    \n
  • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
    • \n
  • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
    • \n
  • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
    • \n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
We need to project the Correlator with a Vector to get a single value at each timeslice.
\n\n
The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
\n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
\n\n
Parameters
\n\n
    \n
  • i (int):\nFirst index to be picked.
    • \n
  • j (int):\nSecond index to be picked.
    • \n
\n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
Outputs the correlator in a plotable format.
\n\n
Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
Symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
Anti-symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
Checks whether a correlator matrices is symmetric on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "
Calculates the per-timeslice trace of a correlator matrix.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\n
The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
\n\n
\n
C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
\n
\n\n
Parameters
\n\n
    \n
  • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
    • \n
  • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
    • \n
  • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
      \n
    • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
      • \n
    • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
      • \n
    • None: The GEVP is solved only at ts, no sorting is necessary
      • \n
    • \n
  • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
    • \n
\n\n
Other Parameters
\n\n
    \n
  • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
    • \n
  • method (str):\nMethod used to solve the GEVP.\n
      \n
    • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
      • \n
    • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
      • \n
    • \n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
Determines the eigenvalue of the GEVP by solving and projecting the correlator
\n\n
Parameters
\n\n
    \n
  • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
    • \n
  • All other parameters are identical to the ones of Corr.GEVP.
    • \n
\n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
Constructs an NxN Hankel matrix
\n\n
C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
\n\n
Parameters
\n\n
    \n
  • N (int):\nDimension of the Hankel matrix
    • \n
  • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
    • \n
\n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
Periodically shift the correlator by dt timeslices
\n\n
Parameters
\n\n
    \n
  • dt (int):\nnumber of timeslices
    • \n
\n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
Reverse the time ordering of the Corr
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
Thin out a correlator to suppress correlations
\n\n
Parameters
\n\n
    \n
  • spacing (int):\nKeep only every 'spacing'th entry of the correlator
    • \n
  • offset (int):\nOffset the equal spacing
    • \n
\n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
Correlate the correlator with another correlator or Obs
\n\n
Parameters
\n\n
    \n
  • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
    • \n
\n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
Reweight the correlator.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
    • \n
\n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
Return the time symmetry average of the correlator and its partner
\n\n
Parameters
\n\n
    \n
  • partner (Corr):\nTime symmetry partner of the Corr
    • \n
  • parity (int):\nParity quantum number of the correlator, can be +1 or -1
    • \n
\n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
Return the first derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
Return the second derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n    - symmetric (default)\n        $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n    - big_symmetric\n        $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n    - improved\n        $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n    - log\n        $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
Returns the effective mass of the correlator as correlator object
\n\n
Parameters
\n\n
    \n
  • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
    • \n
  • guess (float):\nguess for the root finder, only relevant for the root variant
    • \n
\n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
Fits function to the data
\n\n
Parameters
\n\n
    \n
  • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
    • \n
  • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
    • \n
  • silent (bool):\nDecides whether output is printed to the standard output.
    • \n
\n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
Extract a plateau value from a Corr object
\n\n
Parameters
\n\n
    \n
  • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
    • \n
  • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
    • \n
  • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
    • \n
\n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
Sets the attribute prange of the Corr object.
\n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
Plots the correlator using the tag of the correlator as label if available.
\n\n
Parameters
\n\n
    \n
  • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
    • \n
  • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
    • \n
  • logscale (bool):\nSets y-axis to logscale.
    • \n
  • plateau (Obs):\nPlateau value to be visualized in the figure.
    • \n
  • fit_res (Fit_result):\nFit_result object to be visualized.
    • \n
  • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
    • \n
  • ylabel (str):\nLabel for the y-axis.
    • \n
  • save (str):\npath to file in which the figure should be saved.
    • \n
  • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
    • \n
  • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
    • \n
  • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
    • \n
  • title (string):\nOptional title of the figure.
    • \n
\n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\n
Parameters
\n\n
    \n
  • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
    • \n
\n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
Dumps the Corr into a file of chosen type
\n\n
Parameters
\n\n
    \n
  • filename (str):\nName of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
\n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "
\n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
Project large correlation matrix to lowest states
\n\n
This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
\n\n
Parameters
\n\n
    \n
  • Ntrunc (int):\nRank of the target matrix.
    • \n
  • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
    • \n
  • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
    • \n
  • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
    • \n
\n\n
Notes
\n\n
We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
\n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
Initialize Covobs object.
\n\n
Parameters
\n\n
    \n
  • mean (float):\nMean value of the new Obs
    • \n
  • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
Return the variance (= square of the error) of the Covobs
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "
\n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "
\n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
\n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n       [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n       [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "
\n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "
\n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n       [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "
\n", "default_value": "array([[[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n        [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n        [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n        [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],\n        [ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "
\n", "default_value": "array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "
\n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
Rank-3 epsilon tensor
\n\n
Based on https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
    • \n
\n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
Rank-4 epsilon tensor
\n\n
Extension of https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
    • \n
\n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
Returns gamma matrix in Grid labeling.
\n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
\n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
Represents fit results.
\n\n
Attributes
\n\n
    \n
  • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
    • \n
  • chisquare_by_dof (float):\nreduced chisquare.
    • \n
  • p_value (float):\np-value of the fit
    • \n
  • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
    • \n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "
\n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x).\n    ```
\n\n
Parameters
\n\n
    \n
  • For an uncombined fit:
    • \n
  • x (list):\nlist of floats.
    • \n
  • y (list):\nlist of Obs.
    • \n
  • func (object):\nfit function, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • OR For a combined fit:
    • \n
  • x (dict):\ndict of lists.
    • \n
  • y (dict):\ndict of lists of Obs.
    • \n
  • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n        \"b\": func_b}
    \n\n
    def func_a(a, x):\n    return a[1] * anp.exp(-a[0] * x)
    \n\n
    def func_b(a, x):\n    return a[2] * anp.exp(-a[0] * x)
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
    • \n
  • silent (bool, optional):\nIf True all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
    • \n
  • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg\u2013Marquardt is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
    • \n
  • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg\u2013Marquardt) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
    • \n
  • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
    • \n
  • inv_chol_cov_matrix [array,list], optional: array: shape = (number of y values) X (number of y values)\nlist:   for an uncombined fit: [\"\"]\nfor a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order\nIf correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.\nThe matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be\nused to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct\nordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
    • \n
  • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
    • \n
  • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
    • \n
  • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
  • n_parms (int, optional):\nNumber of fit parameters. Overrides automatic detection of parameter count.\nUseful when autodetection fails. Must match the length of initial_guess or priors (if provided).
    • \n
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n\n
Examples
\n\n
\n
>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set\n>>> import numpy as np\n>>> from scipy.stats import norm\n>>> from scipy.linalg import cholesky\n>>> import pyerrors as pe\n>>> # generating the random data set\n>>> num_samples = 400\n>>> N = 3\n>>> x = np.arange(N)\n>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> r = r1 = r2 = np.zeros((N, N))\n>>> y = {}\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix\n>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix\n>>> c = cholesky(r, lower=True)\n>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined\n>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built\n>>> x_dict = {}\n>>> y_dict = {}\n>>> chol_inv_dict = {}\n>>> data = []\n>>> for key in y.keys():\n>>>    x_dict[key] = x\n>>>    for i in range(N):\n>>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data\n>>>    [o.gamma_method() for o in data]\n>>>    corr = pe.covariance(data, correlation=True)\n>>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))\n>>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below\n>>> y_dict = {'a': data[:3], 'b': data[3:]}\n>>> # common fit parameter p[0] in combined fit\n>>> def fit1(p, x):\n>>>    return p[0] + p[1] * x\n>>> def fit2(p, x):\n>>>    return p[0] + p[2] * x\n>>> fitf_dict = {'a': fit1, 'b':fit2}\n>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])\nFit with 3 parameters\nMethod: Levenberg-Marquardt\n`ftol` termination condition is satisfied.\nchisquare/d.o.f.: 0.5388013574561786 # random\nfit parameters [1.11897846 0.96361162 0.92325319] # random\n
\n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\n
Parameters
\n\n
    \n
  • x (list):\nlist of Obs, or a tuple of lists of Obs
    • \n
  • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
    • \n
  • func (object):\nfunc has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • silent (bool, optional):\nIf True all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
    • \n
  • expected_chisquare (bool):\nIf True prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentiation instead of automatic differentiation to perform the error propagation (default False).
    • \n
  • n_parms (int, optional):\nNumber of fit parameters. Overrides automatic detection of parameter count.\nUseful when autodetection fails. Must match the length of initial_guess (if provided).
    • \n
\n\n
Notes
\n\n
Based on the odrpack orthogonal distance regression library.
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
Performs a linear fit to y = n + m * x and returns two Obs n, m.
\n\n
Parameters
\n\n
    \n
  • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
    • \n
  • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
    • \n
\n\n
Returns
\n\n
    \n
  • fit_parameters (list[Obs]):\nLIist of fitted observables.
    • \n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
Generates a plot which compares the fit to the data and displays the corresponding residuals
\n\n
For uncorrelated data the residuals are expected to be distributed ~N(0,1).
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\n
Returns
\n\n
    \n
  • err (np.array(Obs)):\nError band for an array of sample values x
    • \n
\n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\n
Parameters
\n\n
    \n
  • objects (list):\nList of fit results to include in the analysis (optional).
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
\n\n
Jackknife samples
\n\n
For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
\n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
\n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
Extract generic MCMC data from a bdio file
\n\n
read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • data (List[Obs]):\nExtracted data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
Write Obs to a bdio file according to ADerrors conventions
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • success (int):\nreturns 0 is successful
    • \n
\n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
Extract mesons data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n\n
Returns
\n\n
    \n
  • data (dict):\nExtracted meson data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
Extract dSdm data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, kappa)
\n\n
read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
\n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML formatted string of the input data
    • \n
\n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
Import a list of Obs from a string in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • content (str):\nXML string containing the data
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator |is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML string generated from the data
    • \n
\n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
\n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "
Read hadrons hdf5 file and extract entry based on attributes.
\n\n
Parameters
\n\n
    \n
  • filestem (str):\nFull namestem of the files to read, including the full path.
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • group (str):\nlabel of the group to be extracted.
    • \n
  • attrs (dict or int):\nDictionary containing the attributes. For example
    \n\n
    \n
    attrs = {"gamma_snk": "Gamma5",\n         "gamma_src": "Gamma5"}\n
    \n
    \n\n
    Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
Read hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
    • \n
  • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "
Read hadrons FlowObservables hdf5 file and extract t0
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
\n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
Read hadrons DistillationContraction hdf5 files in given directory structure
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the directories to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict):\nextracted DistillationContration data
    • \n
\n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
ndarray(shape, dtype=float, buffer=None, offset=0, strides=None, order=None)
\n\n
An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
\n\n
Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
\n\n
For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
\n\n
Parameters
\n\n
    \n
  • (for the __new__ method; see Notes below)
    • \n
  • shape (tuple of ints):\nShape of created array.
    • \n
  • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.\nDefault is numpy.float64.
    • \n
  • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
    • \n
  • offset (int, optional):\nOffset of array data in buffer.
    • \n
  • strides (tuple of ints, optional):\nStrides of data in memory.
    • \n
  • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
    • \n
\n\n
Attributes
\n\n
    \n
  • T (ndarray):\nTranspose of the array.
    • \n
  • data (buffer):\nThe array's elements, in memory.
    • \n
  • dtype (dtype object):\nDescribes the format of the elements in the array.
    • \n
  • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
    • \n
  • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
    • \n
  • imag (ndarray):\nImaginary part of the array.
    • \n
  • real (ndarray):\nReal part of the array.
    • \n
  • size (int):\nNumber of elements in the array.
    • \n
  • itemsize (int):\nThe memory use of each array element in bytes.
    • \n
  • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
    • \n
  • ndim (int):\nThe array's number of dimensions.
    • \n
  • shape (tuple of ints):\nShape of the array.
    • \n
  • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
    • \n
  • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
    • \n
  • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
    • \n
\n\n
See Also
\n\n
array:  Construct an array.
\nzeros:  Create an array, each element of which is zero.
\nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
\ndtype:  Create a data-type.
\nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
\n\n
Notes
\n\n
There are two modes of creating an array using __new__:
\n\n
    \n
  1. If buffer is None, then only shape, dtype, and order\nare used.
    2. \n
  2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
    3. \n
\n\n
No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
\n\n
Examples
\n\n
These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
\n\n
First mode, buffer is None:
\n\n
\n
>>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
\n
\n\n
Second mode:
\n\n
\n
>>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
\n
\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
Gamma_5 hermitean conjugate
\n\n
Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
\n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
Read hadrons ExternalLeg hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Npr_matrix):\nread Cobs-matrix
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
Read hadrons Bilinear hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict[Npr_matrix]):\nextracted Bilinears
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • vertices (list):\nVertex functions to be extracted.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict):\nextracted fourquark matrizes
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
\n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
\n\n
Returns
\n\n
    \n
  • json_string (str):\nString for export to .json(.gz) file
    • \n
\n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • Null
    • \n
\n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
Reconstruct a list of Obs or structures containing Obs from a json string.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • json_string (str):\njson string containing the data.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
Import a list of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
Export a dict of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
Import a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
    • \n
\n\n
Returns
\n\n
    \n
  • data (Obs / list / Corr):\nRead data
    • \n
  • or
    • \n
  • data (dict):\nRead data and meta-data
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
\n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "
Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
\n\n
It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
\n\n
A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
\n\n
Parameters
\n\n
    \n
  • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
    • \n
  • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
    • \n
\n\n
Returns
\n\n
    \n
  • root (Obs):\nThe root of the data series.
    • \n
\n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
Read pbp format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of observables read
    • \n
\n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
\n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
Read rwms format from given folder structure. Returns a list of length nrw
\n\n
Parameters
\n\n
    \n
  • path (str):\npath that contains the data files
    • \n
  • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
    • \n
  • version (str):\nversion of openQCD, default 2.0
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • print_err (bool):\nPrint additional information that is useful for debugging.
    • \n
\n\n
Returns
\n\n
    \n
  • rwms (Obs):\nReweighting factors read
    • \n
\n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.\nThe function also assumes that r_step is the same across all replica.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • postfix (str):\nPostfix of measurement file (Default: ms)
    • \n
  • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • t0 (Obs):\nExtracted t0
    • \n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "
Extract w0/a from given .ms.dat files. Returns w0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt -  (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • postfix (str):\nPostfix of measurement file (Default: ms)
    • \n
  • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • w0 (Obs):\nExtracted w0
    • \n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
Read the topologial charge based on openQCD gradient flow measurements.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nEither openQCD or sfqcd, depending on the data.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
  • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Obs):\nRead topological charge
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\n
Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
Returns the projection to the topological charge sector defined by target.
\n\n
Parameters
\n\n
    \n
  • path (Obs):\nTopological charge.
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
Constructs reweighting factors to a specified topological sector.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
    • \n
  • r_stop (list):\nlast configurations that need to be read (per replicum)
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
\n\n
Parameters
\n\n
    \n
  • path (str):\nThe directory to search for the files in.
    • \n
  • prefix (str):\nThe prefix to match the files against.
    • \n
  • qc (str):\nThe quark combination extension to match the files against.
    • \n
  • corr (str):\nThe correlator to extract data for.
    • \n
  • sep (str, optional):\nThe separator to use when parsing the replika names.
    • \n
  • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
    \n\n
      \n
    • names (List[str]): A list of names to use for the replicas.
      • \n
    • files (List[str]): A list of files to read data from.
      • \n
    • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
      • \n
    • \n
\n\n
Returns
\n\n
    \n
  • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
    • \n
  • or
    • \n
  • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
    • \n
\n\n
Raises
\n\n
    \n
  • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
    • \n
  • IOError: If there is an error reading a file.
    • \n
  • struct.error: If there is an error unpacking binary data.
    • \n
\n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
\n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
Write DataFrame including Obs or Corr valued columns to sqlite database.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be written to the database.
    • \n
  • table_name (str):\nName of the table in the database.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
    • \n
  • gz (bool):\nIf True the json strings are gzipped.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\n
Parameters
\n\n
    \n
  • sql (str):\nSQL query to be executed.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\n
Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be dumped to a file.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
\n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "
\n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
Read sfcf files from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type (str):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf (int):\nID of wave function
    • \n
  • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
    • \n
\n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tcfg_func=None,\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "
Read sfcf files from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf_list (int):\nID of wave function
    • \n
  • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
  • rep_string (str):\nSeparator of ensemble name and replicum. Example: In \"ensAr0\", \"r\" would be the separator string.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n    dict[key] = list[Obs]\n    where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n    dict[name][quarks][offset][wf][wf2] = list[Obs]
    • \n
\n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tcfg_func=None,\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
Utilities for the input
\n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "
Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.
\n\n
Parameters
\n\n
    \n
  • ll (list):\nlist to sort
    • \n
\n\n
Returns
\n\n
    \n
  • ll (list):\nsorted list
    • \n
\n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
Checks if list of configurations is contained in an idl
\n\n
Parameters
\n\n
    \n
  • idl (range or list):\nidl of the current replicum
    • \n
  • che (list):\nlist of configurations to be checked against
    • \n
\n\n
Returns
\n\n
    \n
  • miss_str (str):\nstring with integers of which idls are missing
    • \n
\n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "
Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
\n\n
Parameters
\n\n
    \n
  • path (str):\nmeasurement path, same as for sfcf read method
    • \n
  • param_hash (str):\nexpected parameter hash
    • \n
  • prefix (str):\ndata prefix to find the appropriate replicum folders in path
    • \n
  • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
    • \n
\n\n
Returns
\n\n
    \n
  • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
    • \n
\n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "
\n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "
Performs a (one-dimensional) numeric integration of f(p, x) from a to b.
\n\n
The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
\n\n
Parameters
\n\n
    \n
  • func (object):\nfunction to integrate, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(p, x):\n    return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
    \n
    \n\n
    where x is the integration variable.
    • \n
  • p (list of floats or Obs):\nparameters of the function func.
    • \n
  • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
    • \n
  • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
    • \n
  • All parameters of scipy.integrate.quad
    • \n
\n\n
Returns
\n\n
    \n
  • y (Obs):\nThe integral of func from a to b.
    • \n
  • abserr (float):\nAn estimate of the absolute error in the result.
    • \n
  • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
    • \n
  • message: A convergence message.
    • \n
  • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
    • \n
\n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
Matrix multiply all operands.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • This implementation is faster compared to standard multiplication via the @ operator.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
Matrix multiply both operands making use of the jackknife approximation.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • For large matrices this is considerably faster compared to matmul.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
Wrapper for numpy.einsum
\n\n
Parameters
\n\n
    \n
  • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
    • \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
Inverse of Obs or CObs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
Cholesky decomposition of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
Determinant of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "
Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
\n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "
Print information about version of python, pyerrors and dependencies.
\n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
pyerrors wrapper for the errorbars method of matplotlib
\n\n
Parameters
\n\n
    \n
  • x (list):\nA list of x-values which can be Obs.
    • \n
  • y (list):\nA list of y-values which can be Obs.
    • \n
  • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
    • \n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.12.13/x64/lib/python3.12/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
Dump object into pickle file.
\n\n
Parameters
\n\n
    \n
  • obj (object):\nobject to be saved in the pickle file
    • \n
  • name (str):\nname of the file
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
Load object from pickle file.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the file
    • \n
\n\n
Returns
\n\n
    \n
  • object (Obs):\nLoaded Object
    • \n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
Generate an Obs object with given value, dvalue and name for test purposes
\n\n
Parameters
\n\n
    \n
  • value (float):\ncentral value of the Obs to be generated.
    • \n
  • dvalue (float):\nerror of the Obs to be generated.
    • \n
  • name (str):\nname of the ensemble for which the Obs is to be generated.
    • \n
  • samples (int):\nnumber of samples for the Obs (default 1000).
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nGenerated Observable
    • \n
\n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
Generate observables with given covariance and autocorrelation times.
\n\n
Parameters
\n\n
    \n
  • means (list):\nlist containing the mean value of each observable.
    • \n
  • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
    • \n
  • name (str):\nensemble name for the data to be geneated.
    • \n
  • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
    • \n
  • samples (int):\nnumber of samples to be generated for each observable.
    • \n
\n\n
Returns
\n\n
    \n
  • corr_obs (list[Obs]):\nGenerated observable list
    • \n
\n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
\n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
Matrix pencil method to extract k energy levels from data
\n\n
Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\n
Parameters
\n\n
    \n
  • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
    • \n
  • k (int):\nNumber of states to extract (default 1).
    • \n
  • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
    • \n
\n\n
Returns
\n\n
    \n
  • energy_levels (list[Obs]):\nExtracted energy levels
    • \n
\n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
\n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
Class for a general observable.
\n\n
Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
\n\n
Attributes
\n\n
    \n
  • S_global (float):\nStandard value for S (default 2.0)
    • \n
  • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
    • \n
  • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
    • \n
  • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
  • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
    • \n
  • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
Initialize Obs object.
\n\n
Parameters
\n\n
    \n
  • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
    • \n
  • names (list):\nlist of strings labeling the individual samples
    • \n
  • idl (list, optional):\nlist of ranges or lists on which the samples are defined
    • \n
\n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "
\n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "
\n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "
\n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "
\n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
Output detailed properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • ens_content (bool):\nprint details about the ensembles and replica if true.
    • \n
\n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
Reweight the obs with given rewighting factors.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
Checks whether the observable is zero within 'sigma' standard errors.
\n\n
Parameters
\n\n
    \n
  • sigma (int):\nNumber of standard errors used for the check.
    • \n
  • Works only properly when the gamma method was run.
    • \n
\n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
Checks whether the observable is zero within a given tolerance.
\n\n
Parameters
\n\n
    \n
  • atol (float):\nAbsolute tolerance (for details see numpy documentation).
    • \n
\n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
Plot integrated autocorrelation time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
Plot normalized autocorrelation function time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
Plot replica distribution for each ensemble with more than one replicum.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
Plot derived Monte Carlo history for each ensemble
\n\n
Parameters
\n\n
    \n
  • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
    • \n
\n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
Dump the Obs to a file 'name' of chosen format.
\n\n
Parameters
\n\n
    \n
  • filename (str):\nname of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • description (str):\nDescription for output file, only relevant for json.gz format.
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
Export jackknife samples from the Obs
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
    • \n
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "
Export bootstrap samples from the Obs
\n\n
Parameters
\n\n
    \n
  • samples (int):\nNumber of bootstrap samples to generate.
    • \n
  • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
    • \n
  • save_rng (str):\nSave the random numbers to a file if a path is specified.
    • \n
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
    • \n
\n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
Class for a complex valued observable.
\n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
\n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "
\n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "
Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\n
See docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "
Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\n
See docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\n
Parameters
\n\n
    \n
  • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
    • \n
  • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
    • \n
  • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
    • \n
\n\n
Notes
\n\n
For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\n
new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
\n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
Reweight a list of observables.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
Correlate two observables.
\n\n
Parameters
\n\n
    \n
  • obs_a (Obs):\nFirst observable
    • \n
  • obs_b (Obs):\nSecond observable
    • \n
\n\n
Notes
\n\n
Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nOnly works if a single ensemble is present in the Obs.\nCurrently only works if ensemble content is identical (this is not strictly necessary).
\n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
Calculates the error covariance matrix of a set of observables.
\n\n
WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
\n\n
The gamma method has to be applied first to all observables.
\n\n
Parameters
\n\n
    \n
  • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
    • \n
  • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
    • \n
  • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
    • \n
  • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
    • \n
\n\n
Notes
\n\n
The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
\n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.invert_corr_cov_cholesky": {"fullname": "pyerrors.obs.invert_corr_cov_cholesky", "modulename": "pyerrors.obs", "qualname": "invert_corr_cov_cholesky", "kind": "function", "doc": "
Constructs a lower triangular matrix chol via the Cholesky decomposition of the correlation matrix corr\n   and then returns the inverse covariance matrix chol_inv as a lower triangular matrix by solving chol * x = inverrdiag.
\n\n
Parameters
\n\n
    \n
  • corr (np.ndarray):\ncorrelation matrix
    • \n
  • inverrdiag (np.ndarray):\ndiagonal matrix, the entries are the inverse errors of the data points considered
    • \n
\n", "signature": "(corr, inverrdiag):", "funcdef": "def"}, "pyerrors.obs.sort_corr": {"fullname": "pyerrors.obs.sort_corr", "modulename": "pyerrors.obs", "qualname": "sort_corr", "kind": "function", "doc": "
Reorders a correlation matrix to match the alphabetical order of its underlying y data.
\n\n
The ordering of the input correlation matrix corr is given by the list of keys kl.\nThe input dictionary yd (with the same keys kl) must contain the corresponding y data\nthat the correlation matrix is based on.\nThis function sorts the list of keys kl alphabetically and sorts the matrix corr\naccording to this alphabetical order such that the sorted matrix corr_sorted corresponds\nto the y data yd when arranged in an alphabetical order by its keys.
\n\n
Parameters
\n\n
    \n
  • corr (np.ndarray):\nA square correlation matrix constructed using the order of the y data specified by kl.\nThe dimensions of corr should match the total number of y data points in yd combined.
    • \n
  • kl (list of str):\nA list of keys that denotes the order in which the y data from yd was used to build the\ninput correlation matrix corr.
    • \n
  • yd (dict of list):\nA dictionary where each key corresponds to a unique identifier, and its value is a list of\ny data points. The total number of y data points across all keys must match the dimensions\nof corr. The lists in the dictionary can be lists of Obs.
    • \n
\n\n
Returns
\n\n
    \n
  • np.ndarray: A new, sorted correlation matrix that corresponds to the y data from yd when arranged alphabetically by its keys.
    • \n
\n\n
Example
\n\n
\n
>>> import numpy as np\n>>> import pyerrors as pe\n>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])\n>>> kl = ['b', 'a']\n>>> yd = {'a': [1, 2], 'b': [3]}\n>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)\n>>> print(sorted_corr)\narray([[1. , 0.3, 0.4],\n       [0.3, 1. , 0.2],\n       [0.4, 0.2, 1. ]])\n
\n
\n", "signature": "(corr, kl, yd):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
Imports jackknife samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
\n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "
Imports bootstrap samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
  • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
    • \n
\n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
Combine all observables in list_of_obs into one new observable.\nThis allows to merge Obs that have been computed on multiple replica\nof the same ensemble.\nIf you like to merge Obs that are based on several ensembles, please\naverage them yourself.
\n\n
Parameters
\n\n
    \n
  • list_of_obs (list):\nlist of the Obs object to be combined
    • \n
\n\n
Notes
\n\n
It is not possible to combine obs which are based on the same replicum
\n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
Create an Obs based on mean(s) and a covariance matrix
\n\n
Parameters
\n\n
    \n
  • mean (list of floats or float):\nN mean value(s) of the new Obs
    • \n
  • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
\n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
Finds the root of the function func(x, d) where d is an Obs.
\n\n
Parameters
\n\n
    \n
  • d (Obs):\nObs passed to the function.
    • \n
  • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
    \n\n
    \n
    import autograd.numpy as anp\ndef root_func(x, d):\n    return anp.exp(-x ** 2) - d\n
    \n
    • \n
  • guess (float):\nInitial guess for the minimization.
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nObs valued root of the function.
    • \n
\n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "
\n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "
beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
beta(a, b, out=None)
\n\n
Beta function.
\n\n
This function is defined in 1 as
\n\n
$$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n        = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
\n\n
where \\( \\Gamma \\) is the gamma function.
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nReal-valued arguments
    • \n
  • out (ndarray, optional):\nOptional output array for the function result
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the beta function
    • \n
\n\n
See Also
\n\n
gamma:  the gamma function
\nbetainc:   the regularized incomplete beta function
\nbetaln:  the natural logarithm of the absolute\nvalue of the beta function  
\n\n
References
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
The beta function relates to the gamma function by the\ndefinition given above:
\n\n
\n
>>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
\n
\n\n
As this relationship demonstrates, the beta function\nis symmetric:
\n\n
\n
>>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
\n
\n\n
This function satisfies \\( B(1, b) = 1/b \\):
\n\n
\n
>>> sc.beta(1, 4)\n0.25\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
betainc(a, b, x, out=None)
\n\n
Regularized incomplete beta function.
\n\n
Computes the regularized incomplete beta function, defined as 1:
\n\n
$$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
\n\n
for \\( 0 \\leq x \\leq 1 \\).
\n\n
This function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nPositive, real-valued parameters
    • \n
  • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the regularized incomplete beta function
    • \n
\n\n
See Also
\n\n
beta()`\nbeta`, `function`  \nbetaincinv()\ninverse, of, the, regularized, incomplete, beta, function
\nbetaincc()`\ncomplement`, `of`, `the`, `regularized`, `incomplete`, `beta`, `function`  \nscipy.stats.beta()\nbeta, distribution  
\n\n
Notes
\n\n
The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula.  When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms.  One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).
\n\n
betainc(a, b, x) is treated as a two parameter family of functions\nof a single variable x, rather than as a function of three variables.\nThis impacts only the limiting cases a = 0, b = 0, a = inf,\nb = inf.
\n\n
In general
\n\n
$$\\lim_{(a, b) \\rightarrow (a_0, b_0)} \\mathrm{betainc}(a, b, x)$$
\n\n
is treated as a pointwise limit in x. Thus for example,\nbetainc(0, b, 0) equals 0 for b > 0, although it would be\nindeterminate when considering the simultaneous limit (a, x) -> (0+, 0+).
\n\n
This function wraps the ibeta routine from the\nBoost Math C++ library 2.
\n\n
Array API Standard Support
\n\n
betainc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Let \\( B(a, b) \\) be the beta function.
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).
\n\n
\n
>>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
\n
\n\n
It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:
\n\n
\n
>>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
\n
\n\n
This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
\n\n
\n
>>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
    \n
    2. \n\n
  2. \n
    The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/. ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
betaln(a, b, out=None)
\n\n
Natural logarithm of absolute value of beta function.
\n\n
Computes ln(abs(beta(a, b))).
\n\n
Parameters
\n\n
    \n
  • a, b (array_like):\nPositive, real-valued parameters
    • \n
  • out (ndarray, optional):\nOptional output array for function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Value of the betaln function
    • \n
\n\n
See Also
\n\n
gamma()`\nthe`, `gamma`, `function`  \nbetainc()\nthe, regularized, incomplete, beta, function
\n`beta()\nthe,beta,function`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
betaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import betaln, beta\n
\n
\n\n
Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):
\n\n
\n
>>> betaln(3, 4)\n-4.0943445622221\n
\n
\n\n
\n
>>> np.log(beta(3, 4))\n-4.0943445622221\n
\n
\n\n
In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.
\n\n
\n
>>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
\n
\n\n
We can compute the logarithm of beta(a, b) by using betaln:
\n\n
\n
>>> betaln(a, b)\n-804.3069951764146\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "
Polygamma functions.
\n\n
Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
    • \n
  • x (array_like):\nReal valued input
    • \n
\n\n
Returns
\n\n
    \n
  • ndarray: Function results
    • \n
\n\n
See Also
\n\n
`digamma()\n..`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
polygamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST, Digital Library of Mathematical Functions,\n    https://dlmf.nist.gov/5.15
\n\n
Examples
\n\n
\n
>>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "
psi(z, out=None)
\n\n
The digamma function.
\n\n
The logarithmic derivative of the gamma function evaluated at z.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex argument.
    • \n
  • out (ndarray, optional):\nArray for the computed values of psi.
    • \n
\n\n
Returns
\n\n
    \n
  • digamma (scalar or ndarray):\nComputed values of psi.
    • \n
\n\n
Notes
\n\n
For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 1 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 1 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 2, so the\nfunction should maintain full accuracy around the origin.
\n\n
Array API Standard Support
\n\n
psi has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n
\n
>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
    \n
    2. \n\n
  2. \n
    Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
psi(z, out=None)
\n\n
The digamma function.
\n\n
The logarithmic derivative of the gamma function evaluated at z.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex argument.
    • \n
  • out (ndarray, optional):\nArray for the computed values of psi.
    • \n
\n\n
Returns
\n\n
    \n
  • digamma (scalar or ndarray):\nComputed values of psi.
    • \n
\n\n
Notes
\n\n
For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 1 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 1 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 2, so the\nfunction should maintain full accuracy around the origin.
\n\n
Array API Standard Support
\n\n
psi has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n
\n
>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n
\n\n
\n
\n
    \n
  1. \n
    NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
    \n
    2. \n\n
  2. \n
    Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
gamma(z, out=None)
\n\n
gamma function.
\n\n
The gamma function is defined as
\n\n
$$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
\n\n
for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex valued argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the gamma function
    • \n
\n\n
Notes
\n\n
The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
\n\n
The gamma function has poles at non-negative integers and the sign\nof infinity as z approaches each pole depends upon the direction in\nwhich the pole is approached. For this reason, the consistent thing\nis for gamma(z) to return NaN at negative integers, and to return\n-inf when x = -0.0 and +inf when x = 0.0, using the signbit of zero\nto signify the direction in which the origin is being approached. This\nis for instance what is recommended for the gamma function in annex F\nentry 9.5.4 of the Iso C 99 standard [isoc99]_.
\n\n
Prior to SciPy version 1.15, scipy.special.gamma(z) returned +inf\nat each pole. This was fixed in version 1.15, but with the following\nconsequence. Expressions where gamma appears in the denominator\nsuch as
\n\n
gamma(u) * gamma(v) / (gamma(w) * gamma(x))
\n\n
no longer evaluate to 0 if the numerator is well defined but there is a\npole in the denominator. Instead such expressions evaluate to NaN. We\nrecommend instead using the function rgamma for the reciprocal gamma\nfunction in such cases. The above expression could for instance be written\nas
\n\n
gamma(u) * gamma(v) * (rgamma(w) * rgamma(x))
\n\n
Array API Standard Support
\n\n
gamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1\n.. [isoc99] https://www.open-std.org/jtc1/sc22/wg14/www/docs/n1256.pdf
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
\n
\n\n
\n
>>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
\n
\n\n
\n
>>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
\n
\n\n
\n
>>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
\n
\n\n
Plot gamma(x) for real x
\n\n
\n
>>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
\n
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "
gammaln(x, out=None)
\n\n
Logarithm of the absolute value of the gamma function.
\n\n
Defined as
\n\n
$$\\ln(\\lvert\\Gamma(x)\\rvert)$$
\n\n
where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the log of the absolute value of gamma
    • \n
\n\n
See Also
\n\n
gammasgn()`\nsign`, `of`, `the`, `gamma`, `function`  \nloggamma()\nprincipal, branch, of, the, logarithm, of, the, gamma, function  
\n\n
Notes
\n\n
It is the same function as the Python standard library function\nmath.lgamma().
\n\n
When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).
\n\n
For complex-valued log-gamma, use loggamma instead of gammaln.
\n\n
Array API Standard Support
\n\n
gammaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import scipy.special as sc\n
\n
\n\n
It has two positive zeros.
\n\n
\n
>>> sc.gammaln([1, 2])\narray([0., 0.])\n
\n
\n\n
It has poles at nonpositive integers.
\n\n
\n
>>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
\n
\n\n
It asymptotically approaches x * log(x) (Stirling's formula).
\n\n
\n
>>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "
gammainc(a, x, out=None)
\n\n
Regularized lower incomplete gamma function.
\n\n
It is defined as
\n\n
$$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
\n\n
for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nPositive parameter
    • \n
  • x (array_like):\nNonnegative argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the lower incomplete gamma function
    • \n
\n\n
See Also
\n\n
gammaincc()`\nregularized`, `upper`, `incomplete`, `gamma`, `function`  \ngammaincinv()\ninverse, of, the, regularized, lower, incomplete, gamma, function
\n`gammainccinv()\ninverse,of,the,regularized,upper,incomplete,gamma,function`  
\n\n
Notes
\n\n
The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.
\n\n
The implementation largely follows that of [boost]_.
\n\n
Array API Standard Support
\n\n
gammainc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
\n\n
\n
>>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
\n
\n\n
It is equal to one minus the upper incomplete gamma function.
\n\n
\n
>>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "
gammaincc(a, x, out=None)
\n\n
Regularized upper incomplete gamma function.
\n\n
It is defined as
\n\n
$$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
\n\n
for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nPositive parameter
    • \n
  • x (array_like):\nNonnegative argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the upper incomplete gamma function
    • \n
\n\n
See Also
\n\n
gammainc()`\nregularized`, `lower`, `incomplete`, `gamma`, `function`  \ngammaincinv()\ninverse, of, the, regularized, lower, incomplete, gamma, function
\n`gammainccinv()\ninverse,of,the,regularized,upper,incomplete,gamma,function`  
\n\n
Notes
\n\n
The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.
\n\n
The implementation largely follows that of [boost]_.
\n\n
Array API Standard Support
\n\n
gammaincc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
\n\n
\n
>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
\n
\n\n
It is equal to one minus the lower incomplete gamma function.
\n\n
\n
>>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "
gammasgn(x, out=None)
\n\n
Sign of the gamma function.
\n\n
It is defined as
\n\n
$$\\text{gammasgn}(x) =\n\\begin{cases}\n  +1 & \\Gamma(x) > 0 \\\n  -1 & \\Gamma(x) < 0\n\\end{cases}$$
\n\n
where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Sign of the gamma function
    • \n
\n\n
See Also
\n\n
gamma()`\nthe`, `gamma`, `function`  \ngammaln()\nlog, of, the, absolute, value, of, the, gamma, function
\n`loggamma()\nanalytic,continuation,of,the,log,of,the,gamma,function`  
\n\n
Notes
\n\n
The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).
\n\n
Array API Standard Support
\n\n
gammasgn has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import scipy.special as sc\n
\n
\n\n
It is 1 for x > 0.
\n\n
\n
>>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
\n
\n\n
It alternates between -1 and 1 for negative integers.
\n\n
\n
>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
\n
\n\n
It can be used to compute the gamma function.
\n\n
\n
>>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "
rgamma(z, out=None)
\n\n
Reciprocal of the gamma function.
\n\n
Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.
\n\n
Parameters
\n\n
    \n
  • z (array_like):\nReal or complex valued input
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Function results
    • \n
\n\n
See Also
\n\n
gamma()`,`,gammaln(),, `loggamma()\n..`  
\n\n
Notes
\n\n
The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.
\n\n
Array API Standard Support
\n\n
rgamma has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
.. [dlmf] Nist, Digital Library of Mathematical functions,\n    https://dlmf.nist.gov/5.2#i
\n\n
Examples
\n\n
\n
>>> import scipy.special as sc\n
\n
\n\n
It is the reciprocal of the gamma function.
\n\n
\n
>>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
\n
\n\n
It is zero at nonpositive integers.
\n\n
\n
>>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
\n
\n\n
It rapidly underflows to zero along the positive real axis.
\n\n
\n
>>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "
Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
\n\n
Parameters
\n\n
    \n
  • a (ndarray):\nThe multivariate gamma is computed for each item of a.
    • \n
  • d (int):\nThe dimension of the space of integration.
    • \n
\n\n
Returns
\n\n
    \n
  • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
    • \n
\n\n
Notes
\n\n
The formal definition of the multivariate gamma of dimension d for a real\na is
\n\n
$$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
\n\n
with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d.  Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).
\n\n
This can be proven to be equal to the much friendlier equation
\n\n
$$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
\n\n
Array API Standard Support
\n\n
multigammaln has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
\n
\n\n
Verify that the result agrees with the logarithm of the equation\nshown above:
\n\n
\n
>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "
Modified Bessel function of the second kind of integer order n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "
j0(x, out=None)
\n\n
Bessel function of the first kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
    • \n
\n\n
See Also
\n\n
jv()`\nBessel`, `function`, `of`, `real`, `order`, `and`, `complex`, `argument.`  \nspherical_jn()\nspherical, Bessel, functions.  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
\n\n
$$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
\n\n
where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
\n\n
In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\n
This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
\n\n
Array API Standard Support
\n\n
j0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
\n
\n\n
Plot the function from -20 to 20.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
y0(x, out=None)
\n\n
Bessel function of the second kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
    • \n
\n\n
See Also
\n\n
j0()`\nBessel`, `function`, `of`, `the`, `first`, `kind`, `of`, `order`, `0`  \nyv()\nBessel, function, of, the, first, kind  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
\n\n
$$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
\n\n
where \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\n
In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\n
This function is a wrapper for the Cephes 1 routine y0.
\n\n
Array API Standard Support
\n\n
y0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
\n
\n\n
Plot the function from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
j1(x, out=None)
\n\n
Bessel function of the first kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
    • \n
\n\n
See Also
\n\n
jv()`\nBessel`, `function`, `of`, `the`, `first`, `kind`  \nspherical_jn()\nspherical, Bessel, functions.  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
\n\n
This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
\n\n
Array API Standard Support
\n\n
j1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
\n
\n\n
Plot the function from -20 to 20.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
y1(x, out=None)
\n\n
Bessel function of the second kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
    • \n
\n\n
See Also
\n\n
j1()`\nBessel`, `function`, `of`, `the`, `first`, `kind`, `of`, `order`, `1`  \nyn()\nBessel, function, of, the, second, kind
\n`yv()\nBessel,function,of,the,second,kind`  
\n\n
Notes
\n\n
The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
\n\n
This function is a wrapper for the Cephes 1 routine y1.
\n\n
Array API Standard Support
\n\n
y1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
\n
\n\n
Plot the function from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
jv(v, z, out=None)
\n\n
Bessel function of the first kind of real order and complex argument.
\n\n
Parameters
\n\n
    \n
  • v (array_like):\nOrder (float).
    • \n
  • z (array_like):\nArgument (float or complex).
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
    • \n
\n\n
See Also
\n\n
jve:  \\( J_v \\) with leading exponential behavior stripped off.
\nspherical_jn:  spherical Bessel functions.
\nj0:  faster version of this function for order 0.
\nj1:  faster version of this function for order 1.  
\n\n
Notes
\n\n
For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),
\n\n
$$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
\n\n
J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
\n\n
For negative v values the formula,
\n\n
$$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
\n\n
is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy.  Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).
\n\n
Not to be confused with the spherical Bessel functions (see spherical_jn).
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -10 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
yn(n, x, out=None)
\n\n
Bessel function of the second kind of integer order and real argument.
\n\n
Parameters
\n\n
    \n
  • n (array_like):\nOrder (integer).
    • \n
  • x (array_like):\nArgument (float).
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
    • \n
\n\n
See Also
\n\n
yv()`\nFor`, `real`, `order`, `and`, `real`, `or`, `complex`, `argument.`  \ny0()\nfaster, implementation, of, this, function, for, order, 0
\n`y1()\nfaster,implementation,of,this,function,for,order,1`  
\n\n
Notes
\n\n
Wrapper for the Cephes 1 routine yn.
\n\n
The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.
\n\n
Array API Standard Support
\n\n
yn has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u26d4
\nJAX                   \u26a0\ufe0f no JIT             \u26d4
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
\n
\n\n
Plot the functions of order 0 to 3 from 0 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
i0(x, out=None)
\n\n
Modified Bessel function of order 0.
\n\n
Defined as,
\n\n
$$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
\n\n
where \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float)
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
    • \n
\n\n
See Also
\n\n
iv()`\nModified`, `Bessel`, `function`, `of`, `any`, `order`  \ni0e()\nExponentially, scaled, modified, Bessel, function, of, order, 0  
\n\n
Notes
\n\n
The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\n
This function is a wrapper for the Cephes 1 routine i0.
\n\n
Array API Standard Support
\n\n
i0 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
\n
\n\n
Calculate at several points:
\n\n
\n
>>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
\n
\n\n
Plot the function from -10 to 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
i1(x, out=None)
\n\n
Modified Bessel function of order 1.
\n\n
Defined as,
\n\n
$$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n       = -\\imath J_1(\\imath x),$$
\n\n
where \\( J_1 \\) is the Bessel function of the first kind of order 1.
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nArgument (float)
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
    • \n
\n\n
See Also
\n\n
iv()`\nModified`, `Bessel`, `function`, `of`, `the`, `first`, `kind`  \ni1e()\nExponentially, scaled, modified, Bessel, function, of, order, 1  
\n\n
Notes
\n\n
The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\n
This function is a wrapper for the Cephes 1 routine i1.
\n\n
Array API Standard Support
\n\n
i1 has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
Calculate the function at one point:
\n\n
\n
>>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
\n
\n\n
Calculate the function at several points:
\n\n
\n
>>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
\n
\n\n
Plot the function between -10 and 10.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
iv(v, z, out=None)
\n\n
Modified Bessel function of the first kind of real order.
\n\n
Parameters
\n\n
    \n
  • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
    • \n
  • z (array_like of float or complex):\nArgument.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the modified Bessel function.
    • \n
\n\n
See Also
\n\n
ive:  This function with leading exponential behavior stripped off.
\ni0:  Faster version of this function for order 0.
\ni1:  Faster version of this function for order 1.  
\n\n
Notes
\n\n
For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1.  For larger orders, uniform asymptotic\nexpansions are applied.
\n\n
For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.
\n\n
The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n
$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n
(valid when the real part of z is positive).  For negative v, the\nformula
\n\n
$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\n
is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
\n\n
References
\n\n
Examples
\n\n
Evaluate the function of order 0 at one point.
\n\n
\n
>>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
\n
\n\n
Evaluate the function at one point for different orders.
\n\n
\n
>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
\n
\n\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
\n
\n\n
If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for a 1D array:
\n\n
\n
>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n
\n\n
\n
>>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩
    \n
    2. \n\n
  2. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    3. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
ive(v, z, out=None)
\n\n
Exponentially scaled modified Bessel function of the first kind.
\n\n
Defined as::
\n\n
ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
\n\n
For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.
\n\n
Parameters
\n\n
    \n
  • v (array_like of float):\nOrder.
    • \n
  • z (array_like of float or complex):\nArgument.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
    • \n
\n\n
See Also
\n\n
iv:  Modified Bessel function of the first kind
\ni0e:  Faster implementation of this function for order 0
\ni1e:  Faster implementation of this function for order 1  
\n\n
Notes
\n\n
For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.
\n\n
The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n
$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n
(valid when the real part of z is positive).  For negative v, the\nformula
\n\n
$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\n
is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
\n\n
ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.
\n\n
References
\n\n
Examples
\n\n
In the following example iv returns infinity whereas ive still returns\na finite number.
\n\n
\n
>>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
\n
\n\n
Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:
\n\n
\n
>>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
\n
\n\n
Evaluate the function at several points for order 0 by providing an\narray for z.
\n\n
\n
>>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
\n
\n\n
Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:
\n\n
\n
>>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
\n
\n\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n
\n
>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
erf(z, out=None)
\n\n
Returns the error function of complex argument.
\n\n
It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nInput array.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • res (scalar or ndarray):\nThe values of the error function at the given points x.
    • \n
\n\n
See Also
\n\n
erfc()`,`,erfinv(),, erfcinv()`,`,wofz(),, erfcx()`,`,erfi()\n..  
\n\n
Notes
\n\n
The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].
\n\n
Array API Standard Support
\n\n
erf has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
erfc(x, out=None)
\n\n
Complementary error function, 1 - erf(x).
\n\n
Parameters
\n\n
    \n
  • x (array_like):\nReal or complex valued argument
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: Values of the complementary error function
    • \n
\n\n
See Also
\n\n
erf()`,`,erfi(),, erfcx()`,`,dawsn(),, `wofz()\n..`  
\n\n
Notes
\n\n
Array API Standard Support
\n\n
erfc has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
erfinv(y, out=None)
\n\n
Inverse of the error function.
\n\n
Computes the inverse of the error function.
\n\n
In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
\n\n
Parameters
\n\n
    \n
  • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
    • \n
\n\n
See Also
\n\n
erf()`\nError`, `function`, `of`, `a`, `complex`, `argument`  \nerfc()\nComplementary, error, function,, 1`, `-`, `erf(x)
\n`erfcinv()\nInverse,of,the,complementary,error,function`  
\n\n
Notes
\n\n
This function wraps the erf_inv routine from the\nBoost Math C++ library 1.
\n\n
Array API Standard Support
\n\n
erfinv has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
References
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
\n
\n\n
\n
>>> erfinv(0.5)\n0.4769362762044699\n
\n
\n\n
\n
>>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
\n
\n\n
Verify that erf(erfinv(y)) is y.
\n\n
\n
>>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
\n
\n\n
Plot the function:
\n\n
\n
>>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
\n
\n\n
\n
\n
    \n
  1. \n
    The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/. ↩
    \n
    2. \n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "
erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n
erfcinv(y, out=None)
\n\n
Inverse of the complementary error function.
\n\n
Computes the inverse of the complementary error function.
\n\n
In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
\n\n
It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
\n\n
Parameters
\n\n
    \n
  • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
    • \n
\n\n
See Also
\n\n
erf:  Error function of a complex argument
\nerfc:  Complementary error function, 1 - erf(x)
\nerfinv:  Inverse of the error function  
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
\n
\n\n
\n
>>> erfcinv(0.5)\n0.4769362762044699\n
\n
\n\n
\n
>>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
\n
\n\n
Plot the function:
\n\n
\n
>>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "
logit(x, out=None)
\n\n
Logit ufunc for ndarrays.
\n\n
The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nThe ndarray to apply logit to element-wise.
    • \n
  • out (ndarray, optional):\nOptional output array for the function results
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
    • \n
\n\n
See Also
\n\n
`expit()\n..`  
\n\n
Notes
\n\n
As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
\n\n
New in version 0.10.0.
\n\n
Array API Standard Support
\n\n
logit has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import logit, expit\n
\n
\n\n
\n
>>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
\n
\n\n
expit is the inverse of logit:
\n\n
\n
>>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
\n
\n\n
Plot logit(x) for x in [0, 1]:
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "
expit(x, out=None)
\n\n
Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.
\n\n
The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)).  It is the inverse of the\nlogit function.
\n\n
Parameters
\n\n
    \n
  • x (ndarray):\nThe ndarray to apply expit to element-wise.
    • \n
  • out (ndarray, optional):\nOptional output array for the function values
    • \n
\n\n
Returns
\n\n
    \n
  • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
    • \n
\n\n
See Also
\n\n
`logit()\n..`  
\n\n
Notes
\n\n
As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
\n\n
New in version 0.10.0.
\n\n
Array API Standard Support
\n\n
expit has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import expit, logit\n
\n
\n\n
\n
>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
\n
\n\n
logit is the inverse of expit:
\n\n
\n
>>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
\n
\n\n
Plot expit(x) for x in [-6, 6]:
\n\n
\n
>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
\n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "
Compute the log of the sum of exponentials of input elements.
\n\n
Parameters
\n\n
    \n
  • a (array_like):\nInput array.
    • \n
  • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.
    \n\n
    New in version 0.11.0.
    • \n
  • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.
    \n\n
    New in version 0.12.0.
    • \n
  • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
    \n\n
    New in version 0.15.0.
    • \n
  • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
    \n\n
    New in version 0.16.0.
    • \n
\n\n
Returns
\n\n
    \n
  • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned. If return_sign is True, res contains the log of\nthe absolute value of the argument.
    • \n
  • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm in res.\nIf return_sign is False, only one result is returned.
    • \n
\n\n
See Also
\n\n
numpy.logaddexp`\n..`  \nnumpy.logaddexp2\n..  
\n\n
Notes
\n\n
NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.
\n\n
The logarithm is a multivalued function: for each \\( x \\) there is an\ninfinite number of \\( z \\) such that \\( exp(z) = x \\). The convention\nis to return the \\( z \\) whose imaginary part lies in \\( (-pi, pi] \\).
\n\n
Array API Standard Support
\n\n
logsumexp has experimental support for Python Array API Standard compatible\nbackends in addition to NumPy. Please consider testing these features\nby setting an environment variable SCIPY_ARRAY_API=1 and providing\nCuPy, PyTorch, JAX, or Dask arrays as array arguments. The following\ncombinations of backend and device (or other capability) are supported.
\n\n
====================  ====================  ====================\nLibrary               CPU                   GPU\n====================  ====================  ====================\nNumPy                 \u2705                     n/a
\nCuPy                  n/a                   \u2705
\nPyTorch               \u2705                     \u2705
\nJAX                   \u2705                     \u2705
\nDask                  \u2705                     n/a
\n====================  ====================  ====================
\n\n
See :ref:`dev-arrayapi` for more information.\n
\n\n
Examples
\n\n
\n
>>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
\n
\n\n
With weights
\n\n
\n
>>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
\n
\n\n
Returning a sign flag
\n\n
\n
>>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
\n
\n\n
Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:
\n\n
\n
>>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
\n
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