{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import pyerrors as pe\n",
"import numpy as np\n",
"import scipy"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As an example we look at a symmetric 2x2 matrix which positive semidefinte and has an error on all entries"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[4.10(20)] Obs[-1.00(10)]]\n",
" [Obs[-1.00(10)] Obs[1.000(10)]]]\n"
]
}
],
"source": [
"obs11 = pe.pseudo_Obs(4.1, 0.2, 'e1')\n",
"obs22 = pe.pseudo_Obs(1, 0.01, 'e1')\n",
"obs12 = pe.pseudo_Obs(-1, 0.1, 'e1')\n",
"matrix = np.asarray([[obs11, obs12], [obs12, obs22]])\n",
"print(matrix)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We require to use `np.asarray` here as it makes sure that we end up with a numpy array of `Obs`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The standard matrix product can be performed with @"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[17.81] Obs[-5.1]]\n",
" [Obs[-5.1] Obs[2.0]]]\n"
]
}
],
"source": [
"print(matrix @ matrix)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Multiplication with unit matrix leaves the matrix unchanged"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[4.1] Obs[-1.0]]\n",
" [Obs[-1.0] Obs[1.0]]]\n"
]
}
],
"source": [
"print(matrix @ np.identity(2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Mathematical functions work elementwise"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[30.161857460980094] Obs[-1.1752011936438014]]\n",
" [Obs[-1.1752011936438014] Obs[1.1752011936438014]]]\n"
]
}
],
"source": [
"print(np.sinh(matrix))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For a vector of `Obs`, we again use np.asarray to end up with the correct object"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Obs[2.00(40)] Obs[1.00(10)]]\n"
]
}
],
"source": [
"vec1 = pe.pseudo_Obs(2, 0.4, 'e1')\n",
"vec2 = pe.pseudo_Obs(1, 0.1, 'e1')\n",
"vector = np.asarray([vec1, vec2])\n",
"for (i), entry in np.ndenumerate(vector):\n",
" entry.gamma_method()\n",
"print(vector)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The matrix times vector product can then be computed via"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Obs[7.2(1.7)] Obs[-1.00(45)]]\n"
]
}
],
"source": [
"product = matrix @ vector\n",
"for (i), entry in np.ndenumerate(product):\n",
" entry.gamma_method()\n",
"print(product)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Matrix to scalar operations\n",
"If we want to apply a numpy matrix function with a scalar return value we can use `scalar_mat_op`. __Here we need to use the autograd wrapped version of numpy__ (imported as anp) to use automatic differentiation."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"det \t Obs[3.10(28)]\n",
"trace \t Obs[5.10(20)]\n",
"norm \t Obs[4.45(19)]\n"
]
}
],
"source": [
"import autograd.numpy as anp # Thinly-wrapped numpy\n",
"funcs = [anp.linalg.det, anp.trace, anp.linalg.norm]\n",
"\n",
"for i, func in enumerate(funcs):\n",
" res = pe.linalg.scalar_mat_op(func, matrix)\n",
" res.gamma_method()\n",
" print(func.__name__, '\\t', res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For matrix operations which are not supported by autograd we can use numerical differentiation"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"cond \t Obs[6.23(58)]\n",
"expm_cond \t Obs[4.45(19)]\n"
]
}
],
"source": [
"funcs = [np.linalg.cond, scipy.linalg.expm_cond]\n",
"\n",
"for i, func in enumerate(funcs):\n",
" res = pe.linalg.scalar_mat_op(func, matrix, num_grad=True)\n",
" res.gamma_method()\n",
" print(func.__name__, ' \\t', res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Matrix to matrix operations\n",
"For matrix operations with a matrix as return value we can use another wrapper `mat_mat_op`. Take as an example the cholesky decompostion. __Here we need to use the autograd wrapped version of numpy__ (imported as anp) to use automatic differentiation."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[2.025(49)] Obs[0.0]]\n",
" [Obs[-0.494(51)] Obs[0.870(29)]]]\n"
]
}
],
"source": [
"cholesky = pe.linalg.mat_mat_op(anp.linalg.cholesky, matrix)\n",
"for (i, j), entry in np.ndenumerate(cholesky):\n",
" entry.gamma_method()\n",
"print(cholesky)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now check if the decomposition was succesfull"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[-8.881784197001252e-16] Obs[0.0]]\n",
" [Obs[0.0] Obs[0.0]]]\n"
]
}
],
"source": [
"check = cholesky @ cholesky.T\n",
"print(check - matrix)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now further compute the inverse of the cholesky decomposed matrix and check that the product with its inverse gives the unit matrix with zero error."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[Obs[0.494(12)] Obs[0.0]]\n",
" [Obs[0.280(40)] Obs[1.150(39)]]]\n",
"Check:\n",
"[[Obs[1.0] Obs[0.0]]\n",
" [Obs[0.0] Obs[1.0]]]\n"
]
}
],
"source": [
"inv = pe.linalg.mat_mat_op(anp.linalg.inv, cholesky)\n",
"for (i, j), entry in np.ndenumerate(inv):\n",
" entry.gamma_method()\n",
"print(inv)\n",
"print('Check:')\n",
"check_inv = cholesky @ inv\n",
"print(check_inv)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Matrix to matrix operations which are not supported by autograd can also be computed with numeric differentiation"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"orth\n",
"[[Obs[-0.9592(76)] Obs[0.283(26)]]\n",
" [Obs[0.283(26)] Obs[0.9592(76)]]]\n",
"expm\n",
"[[Obs[75(15)] Obs[-21.4(4.1)]]\n",
" [Obs[-21.4(4.1)] Obs[8.3(1.4)]]]\n",
"logm\n",
"[[Obs[1.334(57)] Obs[-0.496(61)]]\n",
" [Obs[-0.496(61)] Obs[-0.203(50)]]]\n",
"sinhm\n",
"[[Obs[37.3(7.4)] Obs[-10.8(2.1)]]\n",
" [Obs[-10.8(2.1)] Obs[3.94(68)]]]\n",
"sqrtm\n",
"[[Obs[1.996(51)] Obs[-0.341(37)]]\n",
" [Obs[-0.341(37)] Obs[0.940(14)]]]\n"
]
}
],
"source": [
"funcs = [scipy.linalg.orth, scipy.linalg.expm, scipy.linalg.logm, scipy.linalg.sinhm, scipy.linalg.sqrtm]\n",
"\n",
"for i,func in enumerate(funcs):\n",
" res = pe.linalg.mat_mat_op(func, matrix, num_grad=True)\n",
" for (i, j), entry in np.ndenumerate(res):\n",
" entry.gamma_method()\n",
" print(func.__name__)\n",
" print(res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Eigenvalues and eigenvectors\n",
"We can also compute eigenvalues and eigenvectors of symmetric matrices with a special wrapper `eigh`"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Eigenvalues:\n",
"[Obs[0.705(57)] Obs[4.39(19)]]\n",
"Eigenvectors:\n",
"[[Obs[-0.283(26)] Obs[-0.9592(76)]]\n",
" [Obs[-0.9592(76)] Obs[0.283(26)]]]\n"
]
}
],
"source": [
"e, v = pe.linalg.eigh(matrix)\n",
"for (i), entry in np.ndenumerate(e):\n",
" entry.gamma_method()\n",
"print('Eigenvalues:')\n",
"print(e)\n",
"for (i, j), entry in np.ndenumerate(v):\n",
" entry.gamma_method()\n",
"print('Eigenvectors:')\n",
"print(v)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can check that we got the correct result"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Check eigenvector 1\n",
"[Obs[-5.551115123125783e-17] Obs[0.0]]\n",
"Check eigenvector 2\n",
"[Obs[0.0] Obs[-2.220446049250313e-16]]\n"
]
}
],
"source": [
"for i in range(2):\n",
" print('Check eigenvector', i + 1)\n",
" print(matrix @ v[:, i] - v[:, i] * e[i])"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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