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Initial public release
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435
examples/01_basic_example.ipynb
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examples/01_basic_example.ipynb
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examples/02_pcac_example.ipynb
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examples/02_pcac_example.ipynb
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examples/03_fit_example.ipynb
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examples/03_fit_example.ipynb
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examples/04_matrix_operations.ipynb
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examples/04_matrix_operations.ipynb
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@ -0,0 +1,475 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import sys\n",
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"sys.path.append('..')\n",
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"import pyerrors as pe\n",
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"import numpy as np\n",
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"import scipy"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As an example we look at a symmetric 2x2 matrix which positive semidefinte and has an error on all entries"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[4.10(20)] Obs[-1.00(10)]]\n",
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" [Obs[-1.00(10)] Obs[1.000(10)]]]\n"
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]
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}
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],
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"source": [
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"obs11 = pe.pseudo_Obs(4.1, 0.2, 'e1')\n",
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"obs22 = pe.pseudo_Obs(1, 0.01, 'e1')\n",
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"obs12 = pe.pseudo_Obs(-1, 0.1, 'e1')\n",
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"matrix = np.asarray([[obs11, obs12], [obs12, obs22]])\n",
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"print(matrix)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We require to use `np.asarray` here as it makes sure that we end up with a numpy array of `Obs`."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The standard matrix product can be performed with @"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[17.81] Obs[-5.1]]\n",
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" [Obs[-5.1] Obs[2.0]]]\n"
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]
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}
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],
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"source": [
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"print(matrix @ matrix)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Multiplication with unit matrix leaves the matrix unchanged"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[4.1] Obs[-1.0]]\n",
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" [Obs[-1.0] Obs[1.0]]]\n"
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]
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}
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],
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"source": [
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"print(matrix @ np.identity(2))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Mathematical functions work elementwise"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[30.161857460980094] Obs[-1.1752011936438014]]\n",
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" [Obs[-1.1752011936438014] Obs[1.1752011936438014]]]\n"
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]
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}
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],
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"source": [
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"print(np.sinh(matrix))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"For a vector of `Obs`, we again use np.asarray to end up with the correct object"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Obs[2.00(40)] Obs[1.00(10)]]\n"
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]
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}
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],
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"source": [
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"vec1 = pe.pseudo_Obs(2, 0.4, 'e1')\n",
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"vec2 = pe.pseudo_Obs(1, 0.1, 'e1')\n",
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"vector = np.asarray([vec1, vec2])\n",
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"for (i), entry in np.ndenumerate(vector):\n",
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" entry.gamma_method()\n",
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"print(vector)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The matrix times vector product can then be computed via"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Obs[7.2(1.7)] Obs[-1.00(47)]]\n"
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]
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}
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],
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"source": [
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"product = matrix @ vector\n",
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"for (i), entry in np.ndenumerate(product):\n",
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" entry.gamma_method()\n",
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"print(product)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Matrix to scalar operations\n",
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"det \t Obs[3.10(28)]\n",
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"trace \t Obs[5.10(20)]\n",
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"norm \t Obs[4.45(19)]\n"
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]
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}
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],
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"source": [
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"import autograd.numpy as anp # Thinly-wrapped numpy\n",
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"funcs = [anp.linalg.det, anp.trace, anp.linalg.norm]\n",
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"\n",
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"for i, func in enumerate(funcs):\n",
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" res = pe.linalg.scalar_mat_op(func, matrix)\n",
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" res.gamma_method()\n",
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" print(func.__name__, '\\t', res)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"For matrix operations which are not supported by autograd we can use numerical differentiation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"cond \t Obs[6.23(59)]\n",
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"expm_cond \t Obs[4.45(19)]\n"
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]
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}
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],
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"source": [
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"funcs = [np.linalg.cond, scipy.linalg.expm_cond]\n",
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"\n",
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"for i, func in enumerate(funcs):\n",
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" res = pe.linalg.scalar_mat_op(func, matrix, num_grad=True)\n",
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" res.gamma_method()\n",
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" print(func.__name__, ' \\t', res)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Matrix to matrix operations\n",
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[2.025(49)] Obs[0.0]]\n",
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" [Obs[-0.494(50)] Obs[0.870(29)]]]\n"
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]
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}
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],
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"source": [
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"cholesky = pe.linalg.mat_mat_op(anp.linalg.cholesky, matrix)\n",
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"for (i, j), entry in np.ndenumerate(cholesky):\n",
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" entry.gamma_method()\n",
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"print(cholesky)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We can now check if the decomposition was succesfull"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[-8.881784197001252e-16] Obs[0.0]]\n",
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" [Obs[0.0] Obs[0.0]]]\n"
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]
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}
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],
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"source": [
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"check = cholesky @ cholesky.T\n",
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"print(check - matrix)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[Obs[0.494(12)] Obs[0.0]]\n",
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" [Obs[0.280(40)] Obs[1.150(39)]]]\n",
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"Check:\n",
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"[[Obs[1.0] Obs[0.0]]\n",
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" [Obs[0.0] Obs[1.0]]]\n"
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]
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}
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],
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"source": [
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"inv = pe.linalg.mat_mat_op(anp.linalg.inv, cholesky)\n",
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"for (i, j), entry in np.ndenumerate(inv):\n",
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" entry.gamma_method()\n",
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"print(inv)\n",
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"print('Check:')\n",
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"check_inv = cholesky @ inv\n",
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"print(check_inv)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Matrix to matrix operations which are not supported by autograd can also be computed with numeric differentiation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"orth\n",
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"[[Obs[-0.9592(75)] Obs[0.283(25)]]\n",
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" [Obs[0.283(25)] Obs[0.9592(75)]]]\n",
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"expm\n",
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"[[Obs[75(15)] Obs[-21.4(4.2)]]\n",
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" [Obs[-21.4(4.2)] Obs[8.3(1.4)]]]\n",
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"logm\n",
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"[[Obs[1.334(57)] Obs[-0.496(61)]]\n",
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" [Obs[-0.496(61)] Obs[-0.203(50)]]]\n",
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"sinhm\n",
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"[[Obs[37.3(7.4)] Obs[-10.8(2.1)]]\n",
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" [Obs[-10.8(2.1)] Obs[3.94(69)]]]\n",
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"sqrtm\n",
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"[[Obs[1.996(51)] Obs[-0.341(37)]]\n",
|
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" [Obs[-0.341(37)] Obs[0.940(15)]]]\n"
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]
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}
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||||
],
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"source": [
|
||||
"funcs = [scipy.linalg.orth, scipy.linalg.expm, scipy.linalg.logm, scipy.linalg.sinhm, scipy.linalg.sqrtm]\n",
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"\n",
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||||
"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)"
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||||
]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Eigenvalues and eigenvectors\n",
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||||
"We can also compute eigenvalues and eigenvectors of symmetric matrices with a special wrapper `eigh`"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [
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||||
{
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||||
"name": "stdout",
|
||||
"output_type": "stream",
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||||
"text": [
|
||||
"Eigenvalues:\n",
|
||||
"[Obs[0.705(57)] Obs[4.39(19)]]\n",
|
||||
"Eigenvectors:\n",
|
||||
"[[Obs[-0.283(25)] Obs[-0.9592(75)]]\n",
|
||||
" [Obs[-0.9592(75)] Obs[0.283(25)]]]\n"
|
||||
]
|
||||
}
|
||||
],
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||||
"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": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.6.11"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
BIN
examples/data/B1k2_f_A.p
Normal file
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examples/data/B1k2_f_A.p
Normal file
Binary file not shown.
BIN
examples/data/B1k2_f_P.p
Normal file
BIN
examples/data/B1k2_f_P.p
Normal file
Binary file not shown.
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