pyerrors/examples/05_matrix_operations.ipynb

{
 "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": [
    "For large matrices overloading the standard operator `@` can become inefficient as pyerrors has to perform a large number of elementary opeations. For these situations pyerrors provides the function `linalg.matmul` which optimizes the required automatic differentiation. The function can take an arbitray number of operands."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[Obs[78.12099999999998] Obs[-22.909999999999997]]\n",
      " [Obs[-22.909999999999997] Obs[7.1]]]\n"
     ]
    }
   ],
   "source": [
    "print(pe.linalg.matmul(matrix, matrix, matrix))  # Equivalent to matrix @ matrix @ matrix but faster for large matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mathematical functions work elementwise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": 7,
   "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": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Obs[7.2(1.7)] Obs[-1.00(46)]]\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": [
    "`pyerrors` provides the user with wrappers to the `numpy.linalg` functions which work on `Obs` valued matrices. We can for example calculate the determinant of the matrix via"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.10(28)\n"
     ]
    }
   ],
   "source": [
    "det = pe.linalg.det(matrix)\n",
    "det.gamma_method()\n",
    "print(det)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cholesky decomposition can be obtained as follows"
   ]
  },
  {
   "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(50)] Obs[0.870(29)]]]\n"
     ]
    }
   ],
   "source": [
    "cholesky = pe.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.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": [
    "## Eigenvalues and eigenvectors\n",
    "We can also compute eigenvalues and eigenvectors of symmetric matrices with a special wrapper `eigh`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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(75)]]\n",
      " [Obs[-0.9592(75)] 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": 14,
   "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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