{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear Systems\n", "\n", "Solving linear systems of the form\n", "$$\n", "A \\mathbf{x} = \\mathbf{b}\n", "$$\n", "\n", "where $A$ is symmetric positive definite is arguably one of the most fundamental computations in statistics, machine learning and scientific computation at large. Many problems can be reduced to the solution of one or many (large-scale) linear systems. Some examples include least-squares regression, kernel methods, second-order optimization, quadratic programming, Kalman filtering, linear differential equations and all Gaussian (process) inference. Here, we will solve such a system using a *probabilistic linear solver*." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "pycharm": { "is_executing": false } }, "outputs": [], "source": [ "# Make inline plots vector graphics instead of raster graphics\n", "%matplotlib inline\n", "from IPython.display import set_matplotlib_formats\n", "\n", "set_matplotlib_formats(\"pdf\", \"svg\")\n", "\n", "# Plotting\n", "import matplotlib.pyplot as plt\n", "\n", "plt.style.use(\"../../probnum.mplstyle\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Test Problems\n", "We begin by creating a random linear system with a symmetric positive definite matrix. ProbNum lets you quickly generate test problems via its problem zoo." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ "Matrix condition: 16.00\nEigenvalues: [ 0.625 0.73585981 0.8608519 1.00112915 1.15788966 1.33237674\n 1.52587891 1.73972989 1.97530864 2.23403931 2.51739125 2.82687905\n 3.1640625 3.53054659 3.92798153 4.35806274 4.82253086 5.32317173\n 5.86181641 6.44034115 7.06066744 7.72476196 8.43463662 9.19234853\n 10. ]\n" ] } ], "source": [ "import numpy as np\n", "from probnum.problems.zoo.linalg import random_spd_matrix\n", "\n", "# Random symmetric positive definite linear system\n", "rng = np.random.default_rng(42)\n", "n = 25\n", "\n", "A = random_spd_matrix(\n", " dim=n, spectrum=10 * np.linspace(0.5, 1, n) ** 4, random_state=rng\n", ")\n", "b = rng.normal(size=(n, 1))\n", "\n", "print(\"Matrix condition: {:.2f}\".format(np.linalg.cond(A)))\n", "print(\"Eigenvalues: {}\".format(np.linalg.eigvalsh(A)))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "nbsphinx-thumbnail" ] }, "outputs": [ { "output_type": "display_data", "data": { "text/plain": "