{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "la-0",
   "metadata": {},
   "source": "# Linear algebra\n\nnumbl uses LAPACK for its factorizations (native on the command line,\nWebAssembly in the browser)."
  },
  {
   "cell_type": "markdown",
   "id": "la-1",
   "metadata": {},
   "source": "## Solving a linear system"
  },
  {
   "cell_type": "code",
   "id": "la-2",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "A = [4 -2 1; -2 4 -2; 1 -2 3];\nb = [1; 2; 3];\nx = A \\ b\nresidual = norm(A * x - b)"
  },
  {
   "cell_type": "markdown",
   "id": "la-3",
   "metadata": {},
   "source": "## Determinant, rank, trace, inverse"
  },
  {
   "cell_type": "code",
   "id": "la-4",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "A = [4 -2 1; -2 4 -2; 1 -2 3];\ndet(A)\nrank(A)\ntrace(A)\ninv(A)"
  },
  {
   "cell_type": "markdown",
   "id": "la-5",
   "metadata": {},
   "source": "## Eigenvalues and eigenvectors"
  },
  {
   "cell_type": "code",
   "id": "la-6",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "A = [4 -2 1; -2 4 -2; 1 -2 3];\n[V, D] = eig(A);\neigenvalues = diag(D)'"
  },
  {
   "cell_type": "markdown",
   "id": "la-7",
   "metadata": {},
   "source": "## Singular values and QR"
  },
  {
   "cell_type": "code",
   "id": "la-8",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "A = [4 -2 1; -2 4 -2; 1 -2 3];\ns = svd(A)\n[Q, R] = qr(A);\northogonality_error = norm(Q' * Q - eye(3))"
  },
  {
   "cell_type": "markdown",
   "id": "la-9",
   "metadata": {},
   "source": "## Vector and matrix norms"
  },
  {
   "cell_type": "code",
   "id": "la-10",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "A = [4 -2 1; -2 4 -2; 1 -2 3];\nnorm([3 4])\nnorm([3 4], Inf)\nnorm(A, 'fro')"
  }
 ],
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  "kernelspec": {
   "display_name": "numbl (MATLAB syntax)",
   "language": "numbl",
   "name": "numbl"
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  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "mimetype": "text/x-octave",
   "name": "numbl",
   "nbconvert_exporter": "script",
   "pygments_lexer": "matlab"
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 },
 "nbformat": 4,
 "nbformat_minor": 5
}