# pyerrors.linalg API documentation

# pyerrors.linalg

View Source

      1import numpy as np
      2import autograd.numpy as anp  # Thinly-wrapped numpy
      3from .obs import derived_observable, CObs, Obs, import_jackknife
      4
      5
      6def matmul(*operands):
      7    """Matrix multiply all operands.
      8
      9    Parameters
     10    ----------
     11    operands : numpy.ndarray
     12        Arbitrary number of 2d-numpy arrays which can be real or complex
     13        Obs valued.
     14
     15    This implementation is faster compared to standard multiplication via the @ operator.
     16    """
     17    if any(isinstance(o[0, 0], CObs) for o in operands):
     18        extended_operands = []
     19        for op in operands:
     20            tmp = np.vectorize(lambda x: (np.real(x), np.imag(x)))(op)
     21            extended_operands.append(tmp[0])
     22            extended_operands.append(tmp[1])
     23
     24        def multi_dot(operands, part):
     25            stack_r = operands[0]
     26            stack_i = operands[1]
     27            for op_r, op_i in zip(operands[2::2], operands[3::2]):
     28                tmp_r = stack_r @ op_r - stack_i @ op_i
     29                tmp_i = stack_r @ op_i + stack_i @ op_r
     30
     31                stack_r = tmp_r
     32                stack_i = tmp_i
     33
     34            if part == 'Real':
     35                return stack_r
     36            else:
     37                return stack_i
     38
     39        def multi_dot_r(operands):
     40            return multi_dot(operands, 'Real')
     41
     42        def multi_dot_i(operands):
     43            return multi_dot(operands, 'Imag')
     44
     45        Nr = derived_observable(multi_dot_r, extended_operands, array_mode=True)
     46        Ni = derived_observable(multi_dot_i, extended_operands, array_mode=True)
     47
     48        res = np.empty_like(Nr)
     49        for (n, m), entry in np.ndenumerate(Nr):
     50            res[n, m] = CObs(Nr[n, m], Ni[n, m])
     51
     52        return res
     53    else:
     54        def multi_dot(operands):
     55            stack = operands[0]
     56            for op in operands[1:]:
     57                stack = stack @ op
     58            return stack
     59        return derived_observable(multi_dot, operands, array_mode=True)
     60
     61
     62def jack_matmul(*operands):
     63    """Matrix multiply both operands making use of the jackknife approximation.
     64
     65    Parameters
     66    ----------
     67    operands : numpy.ndarray
     68        Arbitrary number of 2d-numpy arrays which can be real or complex
     69        Obs valued.
     70
     71    For large matrices this is considerably faster compared to matmul.
     72    """
     73
     74    def _exp_to_jack(matrix):
     75        base_matrix = np.empty_like(matrix)
     76        for index, entry in np.ndenumerate(matrix):
     77            base_matrix[index] = entry.export_jackknife()
     78        return base_matrix
     79
     80    def _imp_from_jack(matrix, name, idl):
     81        base_matrix = np.empty_like(matrix)
     82        for index, entry in np.ndenumerate(matrix):
     83            base_matrix[index] = import_jackknife(entry, name, [idl])
     84        return base_matrix
     85
     86    def _exp_to_jack_c(matrix):
     87        base_matrix = np.empty_like(matrix)
     88        for index, entry in np.ndenumerate(matrix):
     89            base_matrix[index] = entry.real.export_jackknife() + 1j * entry.imag.export_jackknife()
     90        return base_matrix
     91
     92    def _imp_from_jack_c(matrix, name, idl):
     93        base_matrix = np.empty_like(matrix)
     94        for index, entry in np.ndenumerate(matrix):
     95            base_matrix[index] = CObs(import_jackknife(entry.real, name, [idl]),
     96                                      import_jackknife(entry.imag, name, [idl]))
     97        return base_matrix
     98
     99    if any(isinstance(o.flat[0], CObs) for o in operands):
    100        name = operands[0].flat[0].real.names[0]
    101        idl = operands[0].flat[0].real.idl[name]
    102
    103        r = _exp_to_jack_c(operands[0])
    104        for op in operands[1:]:
    105            if isinstance(op.flat[0], CObs):
    106                r = r @ _exp_to_jack_c(op)
    107            else:
    108                r = r @ op
    109        return _imp_from_jack_c(r, name, idl)
    110    else:
    111        name = operands[0].flat[0].names[0]
    112        idl = operands[0].flat[0].idl[name]
    113
    114        r = _exp_to_jack(operands[0])
    115        for op in operands[1:]:
    116            if isinstance(op.flat[0], Obs):
    117                r = r @ _exp_to_jack(op)
    118            else:
    119                r = r @ op
    120        return _imp_from_jack(r, name, idl)
    121
    122
    123def einsum(subscripts, *operands):
    124    """Wrapper for numpy.einsum
    125
    126    Parameters
    127    ----------
    128    subscripts : str
    129        Subscripts for summation (see numpy documentation for details)
    130    operands : numpy.ndarray
    131        Arbitrary number of 2d-numpy arrays which can be real or complex
    132        Obs valued.
    133    """
    134
    135    def _exp_to_jack(matrix):
    136        base_matrix = []
    137        for index, entry in np.ndenumerate(matrix):
    138            base_matrix.append(entry.export_jackknife())
    139        return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)
    140
    141    def _exp_to_jack_c(matrix):
    142        base_matrix = []
    143        for index, entry in np.ndenumerate(matrix):
    144            base_matrix.append(entry.real.export_jackknife() + 1j * entry.imag.export_jackknife())
    145        return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)
    146
    147    def _imp_from_jack(matrix, name, idl):
    148        base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
    149        for index in np.ndindex(matrix.shape[:-1]):
    150            base_matrix[index] = import_jackknife(matrix[index], name, [idl])
    151        return base_matrix
    152
    153    def _imp_from_jack_c(matrix, name, idl):
    154        base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
    155        for index in np.ndindex(matrix.shape[:-1]):
    156            base_matrix[index] = CObs(import_jackknife(matrix[index].real, name, [idl]),
    157                                      import_jackknife(matrix[index].imag, name, [idl]))
    158        return base_matrix
    159
    160    for op in operands:
    161        if isinstance(op.flat[0], CObs):
    162            name = op.flat[0].real.names[0]
    163            idl = op.flat[0].real.idl[name]
    164            break
    165        elif isinstance(op.flat[0], Obs):
    166            name = op.flat[0].names[0]
    167            idl = op.flat[0].idl[name]
    168            break
    169
    170    conv_operands = []
    171    for op in operands:
    172        if isinstance(op.flat[0], CObs):
    173            conv_operands.append(_exp_to_jack_c(op))
    174        elif isinstance(op.flat[0], Obs):
    175            conv_operands.append(_exp_to_jack(op))
    176        else:
    177            conv_operands.append(op)
    178
    179    tmp_subscripts = ','.join([o + '...' for o in subscripts.split(',')])
    180    extended_subscripts = '->'.join([o + '...' for o in tmp_subscripts.split('->')[:-1]] + [tmp_subscripts.split('->')[-1]])
    181    einsum_path = np.einsum_path(extended_subscripts, *conv_operands, optimize='optimal')[0]
    182    jack_einsum = np.einsum(extended_subscripts, *conv_operands, optimize=einsum_path)
    183
    184    if jack_einsum.dtype == complex:
    185        result = _imp_from_jack_c(jack_einsum, name, idl)
    186    elif jack_einsum.dtype == float:
    187        result = _imp_from_jack(jack_einsum, name, idl)
    188    else:
    189        raise Exception("Result has unexpected datatype")
    190
    191    if result.shape == ():
    192        return result.flat[0]
    193    else:
    194        return result
    195
    196
    197def inv(x):
    198    """Inverse of Obs or CObs valued matrices."""
    199    return _mat_mat_op(anp.linalg.inv, x)
    200
    201
    202def cholesky(x):
    203    """Cholesky decomposition of Obs valued matrices."""
    204    if any(isinstance(o, CObs) for o in x.ravel()):
    205        raise Exception("Cholesky decomposition is not implemented for CObs.")
    206    return _mat_mat_op(anp.linalg.cholesky, x)
    207
    208
    209def det(x):
    210    """Determinant of Obs valued matrices."""
    211    return _scalar_mat_op(anp.linalg.det, x)
    212
    213
    214def _scalar_mat_op(op, obs, **kwargs):
    215    """Computes the matrix to scalar operation op to a given matrix of Obs."""
    216    def _mat(x, **kwargs):
    217        dim = int(np.sqrt(len(x)))
    218
    219        mat = []
    220        for i in range(dim):
    221            row = []
    222            for j in range(dim):
    223                row.append(x[j + dim * i])
    224            mat.append(row)
    225
    226        return op(anp.array(mat))
    227
    228    if isinstance(obs, np.ndarray):
    229        raveled_obs = (1 * (obs.ravel())).tolist()
    230    else:
    231        raise TypeError('Unproper type of input.')
    232    return derived_observable(_mat, raveled_obs, **kwargs)
    233
    234
    235def _mat_mat_op(op, obs, **kwargs):
    236    """Computes the matrix to matrix operation op to a given matrix of Obs."""
    237    # Use real representation to calculate matrix operations for complex matrices
    238    if any(isinstance(o, CObs) for o in obs.ravel()):
    239        A = np.empty_like(obs)
    240        B = np.empty_like(obs)
    241        for (n, m), entry in np.ndenumerate(obs):
    242            if hasattr(entry, 'real') and hasattr(entry, 'imag'):
    243                A[n, m] = entry.real
    244                B[n, m] = entry.imag
    245            else:
    246                A[n, m] = entry
    247                B[n, m] = 0.0
    248        big_matrix = np.block([[A, -B], [B, A]])
    249        op_big_matrix = derived_observable(lambda x, **kwargs: op(x), [big_matrix], array_mode=True)[0]
    250        dim = op_big_matrix.shape[0]
    251        op_A = op_big_matrix[0: dim // 2, 0: dim // 2]
    252        op_B = op_big_matrix[dim // 2:, 0: dim // 2]
    253        res = np.empty_like(op_A)
    254        for (n, m), entry in np.ndenumerate(op_A):
    255            res[n, m] = CObs(op_A[n, m], op_B[n, m])
    256        return res
    257    else:
    258        return derived_observable(lambda x, **kwargs: op(x), [obs], array_mode=True)[0]
    259
    260
    261def eigh(obs, **kwargs):
    262    """Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
    263    w = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[0], obs)
    264    v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
    265    return w, v
    266
    267
    268def eig(obs, **kwargs):
    269    """Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig."""
    270    w = derived_observable(lambda x, **kwargs: anp.real(anp.linalg.eig(x)[0]), obs)
    271    return w
    272
    273
    274def eigv(obs, **kwargs):
    275    """Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
    276    v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
    277    return v
    278
    279
    280def pinv(obs, **kwargs):
    281    """Computes the Moore-Penrose pseudoinverse of a matrix of Obs."""
    282    return derived_observable(lambda x, **kwargs: anp.linalg.pinv(x), obs)
    283
    284
    285def svd(obs, **kwargs):
    286    """Computes the singular value decomposition of a matrix of Obs."""
    287    u = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[0], obs)
    288    s = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[1], obs)
    289    vh = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[2], obs)
    290    return (u, s, vh)

def matmul(*operands): View Source

     7def matmul(*operands):
     8    """Matrix multiply all operands.
     9
    10    Parameters
    11    ----------
    12    operands : numpy.ndarray
    13        Arbitrary number of 2d-numpy arrays which can be real or complex
    14        Obs valued.
    15
    16    This implementation is faster compared to standard multiplication via the @ operator.
    17    """
    18    if any(isinstance(o[0, 0], CObs) for o in operands):
    19        extended_operands = []
    20        for op in operands:
    21            tmp = np.vectorize(lambda x: (np.real(x), np.imag(x)))(op)
    22            extended_operands.append(tmp[0])
    23            extended_operands.append(tmp[1])
    24
    25        def multi_dot(operands, part):
    26            stack_r = operands[0]
    27            stack_i = operands[1]
    28            for op_r, op_i in zip(operands[2::2], operands[3::2]):
    29                tmp_r = stack_r @ op_r - stack_i @ op_i
    30                tmp_i = stack_r @ op_i + stack_i @ op_r
    31
    32                stack_r = tmp_r
    33                stack_i = tmp_i
    34
    35            if part == 'Real':
    36                return stack_r
    37            else:
    38                return stack_i
    39
    40        def multi_dot_r(operands):
    41            return multi_dot(operands, 'Real')
    42
    43        def multi_dot_i(operands):
    44            return multi_dot(operands, 'Imag')
    45
    46        Nr = derived_observable(multi_dot_r, extended_operands, array_mode=True)
    47        Ni = derived_observable(multi_dot_i, extended_operands, array_mode=True)
    48
    49        res = np.empty_like(Nr)
    50        for (n, m), entry in np.ndenumerate(Nr):
    51            res[n, m] = CObs(Nr[n, m], Ni[n, m])
    52
    53        return res
    54    else:
    55        def multi_dot(operands):
    56            stack = operands[0]
    57            for op in operands[1:]:
    58                stack = stack @ op
    59            return stack
    60        return derived_observable(multi_dot, operands, array_mode=True)

Matrix multiply all operands.

###### Parameters

- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.
- This implementation is faster compared to standard multiplication via the @ operator.

def jack_matmul(*operands): View Source

     63def jack_matmul(*operands):
     64    """Matrix multiply both operands making use of the jackknife approximation.
     65
     66    Parameters
     67    ----------
     68    operands : numpy.ndarray
     69        Arbitrary number of 2d-numpy arrays which can be real or complex
     70        Obs valued.
     71
     72    For large matrices this is considerably faster compared to matmul.
     73    """
     74
     75    def _exp_to_jack(matrix):
     76        base_matrix = np.empty_like(matrix)
     77        for index, entry in np.ndenumerate(matrix):
     78            base_matrix[index] = entry.export_jackknife()
     79        return base_matrix
     80
     81    def _imp_from_jack(matrix, name, idl):
     82        base_matrix = np.empty_like(matrix)
     83        for index, entry in np.ndenumerate(matrix):
     84            base_matrix[index] = import_jackknife(entry, name, [idl])
     85        return base_matrix
     86
     87    def _exp_to_jack_c(matrix):
     88        base_matrix = np.empty_like(matrix)
     89        for index, entry in np.ndenumerate(matrix):
     90            base_matrix[index] = entry.real.export_jackknife() + 1j * entry.imag.export_jackknife()
     91        return base_matrix
     92
     93    def _imp_from_jack_c(matrix, name, idl):
     94        base_matrix = np.empty_like(matrix)
     95        for index, entry in np.ndenumerate(matrix):
     96            base_matrix[index] = CObs(import_jackknife(entry.real, name, [idl]),
     97                                      import_jackknife(entry.imag, name, [idl]))
     98        return base_matrix
     99
    100    if any(isinstance(o.flat[0], CObs) for o in operands):
    101        name = operands[0].flat[0].real.names[0]
    102        idl = operands[0].flat[0].real.idl[name]
    103
    104        r = _exp_to_jack_c(operands[0])
    105        for op in operands[1:]:
    106            if isinstance(op.flat[0], CObs):
    107                r = r @ _exp_to_jack_c(op)
    108            else:
    109                r = r @ op
    110        return _imp_from_jack_c(r, name, idl)
    111    else:
    112        name = operands[0].flat[0].names[0]
    113        idl = operands[0].flat[0].idl[name]
    114
    115        r = _exp_to_jack(operands[0])
    116        for op in operands[1:]:
    117            if isinstance(op.flat[0], Obs):
    118                r = r @ _exp_to_jack(op)
    119            else:
    120                r = r @ op
    121        return _imp_from_jack(r, name, idl)

Matrix multiply both operands making use of the jackknife approximation.

###### Parameters

- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.
- For large matrices this is considerably faster compared to matmul.

def einsum(subscripts, *operands): View Source

    124def einsum(subscripts, *operands):
    125    """Wrapper for numpy.einsum
    126
    127    Parameters
    128    ----------
    129    subscripts : str
    130        Subscripts for summation (see numpy documentation for details)
    131    operands : numpy.ndarray
    132        Arbitrary number of 2d-numpy arrays which can be real or complex
    133        Obs valued.
    134    """
    135
    136    def _exp_to_jack(matrix):
    137        base_matrix = []
    138        for index, entry in np.ndenumerate(matrix):
    139            base_matrix.append(entry.export_jackknife())
    140        return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)
    141
    142    def _exp_to_jack_c(matrix):
    143        base_matrix = []
    144        for index, entry in np.ndenumerate(matrix):
    145            base_matrix.append(entry.real.export_jackknife() + 1j * entry.imag.export_jackknife())
    146        return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)
    147
    148    def _imp_from_jack(matrix, name, idl):
    149        base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
    150        for index in np.ndindex(matrix.shape[:-1]):
    151            base_matrix[index] = import_jackknife(matrix[index], name, [idl])
    152        return base_matrix
    153
    154    def _imp_from_jack_c(matrix, name, idl):
    155        base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
    156        for index in np.ndindex(matrix.shape[:-1]):
    157            base_matrix[index] = CObs(import_jackknife(matrix[index].real, name, [idl]),
    158                                      import_jackknife(matrix[index].imag, name, [idl]))
    159        return base_matrix
    160
    161    for op in operands:
    162        if isinstance(op.flat[0], CObs):
    163            name = op.flat[0].real.names[0]
    164            idl = op.flat[0].real.idl[name]
    165            break
    166        elif isinstance(op.flat[0], Obs):
    167            name = op.flat[0].names[0]
    168            idl = op.flat[0].idl[name]
    169            break
    170
    171    conv_operands = []
    172    for op in operands:
    173        if isinstance(op.flat[0], CObs):
    174            conv_operands.append(_exp_to_jack_c(op))
    175        elif isinstance(op.flat[0], Obs):
    176            conv_operands.append(_exp_to_jack(op))
    177        else:
    178            conv_operands.append(op)
    179
    180    tmp_subscripts = ','.join([o + '...' for o in subscripts.split(',')])
    181    extended_subscripts = '->'.join([o + '...' for o in tmp_subscripts.split('->')[:-1]] + [tmp_subscripts.split('->')[-1]])
    182    einsum_path = np.einsum_path(extended_subscripts, *conv_operands, optimize='optimal')[0]
    183    jack_einsum = np.einsum(extended_subscripts, *conv_operands, optimize=einsum_path)
    184
    185    if jack_einsum.dtype == complex:
    186        result = _imp_from_jack_c(jack_einsum, name, idl)
    187    elif jack_einsum.dtype == float:
    188        result = _imp_from_jack(jack_einsum, name, idl)
    189    else:
    190        raise Exception("Result has unexpected datatype")
    191
    192    if result.shape == ():
    193        return result.flat[0]
    194    else:
    195        return result

Wrapper for numpy.einsum

###### Parameters

- subscripts (str): Subscripts for summation (see numpy documentation for details)
- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.

def inv(x): View Source

    198def inv(x):
    199    """Inverse of Obs or CObs valued matrices."""
    200    return _mat_mat_op(anp.linalg.inv, x)

Inverse of Obs or CObs valued matrices.

def cholesky(x): View Source

    203def cholesky(x):
    204    """Cholesky decomposition of Obs valued matrices."""
    205    if any(isinstance(o, CObs) for o in x.ravel()):
    206        raise Exception("Cholesky decomposition is not implemented for CObs.")
    207    return _mat_mat_op(anp.linalg.cholesky, x)

Cholesky decomposition of Obs valued matrices.

def det(x): View Source

    210def det(x):
    211    """Determinant of Obs valued matrices."""
    212    return _scalar_mat_op(anp.linalg.det, x)

Determinant of Obs valued matrices.

def eigh(obs, **kwargs): View Source

    262def eigh(obs, **kwargs):
    263    """Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
    264    w = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[0], obs)
    265    v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
    266    return w, v

Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

def eig(obs, **kwargs): View Source

    269def eig(obs, **kwargs):
    270    """Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig."""
    271    w = derived_observable(lambda x, **kwargs: anp.real(anp.linalg.eig(x)[0]), obs)
    272    return w

Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.

def eigv(obs, **kwargs): View Source

    275def eigv(obs, **kwargs):
    276    """Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
    277    v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
    278    return v

Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

def pinv(obs, **kwargs): View Source

    281def pinv(obs, **kwargs):
    282    """Computes the Moore-Penrose pseudoinverse of a matrix of Obs."""
    283    return derived_observable(lambda x, **kwargs: anp.linalg.pinv(x), obs)

Computes the Moore-Penrose pseudoinverse of a matrix of Obs.

def svd(obs, **kwargs): View Source

    286def svd(obs, **kwargs):
    287    """Computes the singular value decomposition of a matrix of Obs."""
    288    u = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[0], obs)
    289    s = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[1], obs)
    290    vh = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[2], obs)
    291    return (u, s, vh)

Computes the singular value decomposition of a matrix of Obs.