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Documenter.jl
163
dev/notebooks/LEQ.ipynb
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163
dev/notebooks/LEQ.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "599913a3",
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"metadata": {},
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"source": [
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"# Solving Linear Equations\n",
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"\n",
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"## Serial Algorithm\n",
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"First, we construct a linear equations system $Ax=b$."
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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": null,
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"id": "fe3b5b02",
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"metadata": {},
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"outputs": [],
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"source": [
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"n = 5\n",
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"A = rand(-10.0:10.0, (n, n))"
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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": null,
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"id": "57c912fb",
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"metadata": {},
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"outputs": [],
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"source": [
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"x = rand(-10.0:10.0, n)\n",
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"b = A * x"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4baa0681",
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"metadata": {},
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"source": [
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"The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$. "
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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": null,
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"id": "11d255e0",
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"metadata": {},
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"outputs": [],
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"source": [
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"function convert_to_upper_triangular(A,b)\n",
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" # Upper Triangularization: convert Ax=b to Ux=y\n",
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" for k in 1:n\n",
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" for j in k+1:n\n",
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" # Divide by pivot\n",
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" A[k,j] = A[k,j] / A[k,k]\n",
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" end\n",
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" b[k] = b[k] / A[k,k]\n",
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" A[k,k] = 1\n",
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" # Substract lower rows\n",
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" for i in k+1:n \n",
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" for j in k+1:n\n",
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" A[i,j]=A[i,j] - A[i,k] * A[k,j]\n",
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" end\n",
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" b[i] = b[i] - A[i,k] * b[k]\n",
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" A[i,k] = 0\n",
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" end\n",
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" end\n",
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" return A, b #U,y\n",
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"end\n"
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]
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},
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{
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||||
"cell_type": "markdown",
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"id": "02c47593",
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"metadata": {},
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"source": [
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"The function in the following cell solves the upper triangular equation system using backwards substitution. "
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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": null,
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"id": "3c19497c",
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"metadata": {},
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"outputs": [],
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"source": [
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"function solve_upper_triangular(U,y)\n",
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" n = size(U,1)\n",
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" for step in reverse(1:n)\n",
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" if U[step,step] == 0\n",
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" if y[step] != 0\n",
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" return \"No solution\"\n",
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" else\n",
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" return \"Infinity solutions\"\n",
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" end\n",
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" else\n",
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" # Backwards substitution\n",
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" y[step] = y[step] / U[step,step]\n",
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" end\n",
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" for row in reverse(1:step-1)\n",
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||||
" y[row] -= U[row,step] * y[step]\n",
|
||||
" end\n",
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||||
" end\n",
|
||||
" return y \n",
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||||
"end"
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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": null,
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"id": "2c24d85e",
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"metadata": {},
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"outputs": [],
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"source": [
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"U,y = convert_to_upper_triangular(A,b)\n",
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"sol = solve_upper_triangular(U,y)"
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]
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},
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||||
{
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||||
"cell_type": "markdown",
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"id": "1c356b5a",
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"metadata": {},
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"source": [
|
||||
"We can test if the obtained solution is correct using `@test`:"
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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": null,
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"id": "5c71828d",
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"metadata": {},
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"outputs": [],
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"source": [
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"using Test\n",
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"@test sol ≈ x"
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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": null,
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"id": "fe3c5374",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Julia 1.9.1",
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"language": "julia",
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"name": "julia-1.9"
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},
|
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.9.1"
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}
|
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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21
dev/notebooks/LEQ/index.html
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dev/notebooks/LEQ/index.html
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<!DOCTYPE html>
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>- · XM_40017</title><script data-outdated-warner src="../../assets/warner.js"></script><link rel="canonical" href="https://fverdugo.github.io/XM_40017/notebooks/LEQ/"/><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/><script src="../../assets/iframeResizer.min.js"></script><link href="../../assets/custom.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="XM_40017 logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../../">XM_40017</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../getting_started_with_julia/">Getting started</a></li><li><span class="tocitem">Notebooks</span><ul><li><a class="tocitem" href="../julia_basics/">Julia Basics</a></li><li><a class="tocitem" href="../julia_async/">Tasks and channels</a></li><li><a class="tocitem" href="../julia_distributed/">Remote calls and remote channels</a></li><li><a class="tocitem" href="../mpi_tutorial/">MPI</a></li><li><a class="tocitem" href="../matrix_matrix/">Matrix Multiplication</a></li><li><a class="tocitem" href="../jacobi_method/">Jacobi</a></li><li><a class="tocitem" href="../asp/">ASP</a></li><li><a class="tocitem" href="../solutions/">Solutions</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>-</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>-</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/fverdugo/XM_40017/blob/main/docs/src/notebooks/LEQ.ipynb" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><div class="admonition is-success">
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<header class="admonition-header">Tip</header>
|
||||
<div class="admonition-body">
|
||||
<ul>
|
||||
<li>
|
||||
Download this notebook and run it locally on your machine [recommended]. Click <a href="LEQ.ipynb" download>here</a>.
|
||||
</li>
|
||||
<li>
|
||||
You can also run this notebook in the cloud using Binder. Click <a href="https://mybinder.org/v2/gh/fverdugo/XM_40017/gh-pages?filepath=dev/notebooks/LEQ.ipynb">here</a>
|
||||
.
|
||||
</li>
|
||||
</ul>
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||||
</div>
|
||||
</div><iframe id="notebook" src="../notebook-html/LEQ.html" style="width:100%"></iframe>
|
||||
<script>
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document.addEventListener('DOMContentLoaded', function(){
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var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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});
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</script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 22 August 2023 10:48">Tuesday 22 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -18,4 +18,4 @@
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var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 21 August 2023 14:09">Monday 21 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 22 August 2023 10:48">Tuesday 22 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -18,4 +18,4 @@
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var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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});
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</script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 21 August 2023 14:09">Monday 21 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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</script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 22 August 2023 10:48">Tuesday 22 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
@@ -18,4 +18,4 @@
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var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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||||
});
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../matrix_matrix/">« Matrix Multiplication</a><a class="docs-footer-nextpage" href="../asp/">ASP »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 21 August 2023 14:09">Monday 21 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../matrix_matrix/">« Matrix Multiplication</a><a class="docs-footer-nextpage" href="../asp/">ASP »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 22 August 2023 10:48">Tuesday 22 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -18,4 +18,4 @@
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var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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||||
});
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../julia_basics/">« Julia Basics</a><a class="docs-footer-nextpage" href="../julia_distributed/">Remote calls and remote channels »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 21 August 2023 14:09">Monday 21 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../julia_basics/">« Julia Basics</a><a class="docs-footer-nextpage" href="../julia_distributed/">Remote calls and remote channels »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 22 August 2023 10:48">Tuesday 22 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
@@ -18,4 +18,4 @@
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||||
var myIframe = document.getElementById("notebook");
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iFrameResize({log:true}, myIframe);
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||||
});
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@@ -18,4 +18,4 @@
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@@ -18,4 +18,4 @@
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@@ -18,4 +18,4 @@
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@@ -18,4 +18,4 @@
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@@ -18,4 +18,4 @@
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@@ -18,4 +18,4 @@
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Reference in New Issue
Block a user