mirror of
https://github.com/fverdugo/XM_40017.git
synced 2025-12-30 02:38:31 +01:00
build based on 1f375dd
This commit is contained in:
@@ -2,53 +2,59 @@
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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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"id": "9c32b051",
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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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"To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:"
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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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"execution_count": 66,
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"id": "2b369b73",
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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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"A = [1.0 4.0 5.0 8.0 1.0; \n",
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" 2.0 -1.0 4.0 3.0 0.0; \n",
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" 7.0 6.0 3.0 -4.0 5.0; \n",
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" -3.0 4.0 2.0 2.0 2.0; \n",
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" 0.0 -4.0 2.0 1.0 2.0]\n",
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"\n",
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"b = [61.0; 24.0; 37.0; 29.0; 12.0];"
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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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"id": "53124eb8",
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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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"The code in the following cell converts the general problem $Ax=b$ to the upper triangular equation system $Ux=y$. Note that this function assumes that the pivots are all nonzero. This function will be erroneos if any of the diagonal entries are zero!"
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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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"execution_count": 67,
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"id": "7a7b926a",
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||||
"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"text/plain": [
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"convert_to_upper_triangular! (generic function with 1 method)"
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]
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},
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"execution_count": 67,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"function convert_to_upper_triangular(A,b)\n",
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"function convert_to_upper_triangular!(A,b)\n",
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" n = size(A,1)\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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@@ -72,20 +78,31 @@
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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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"id": "78ef1849",
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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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"The function in the following cell solves the upper triangular equation system using backwards substitution. Note that the function alters the input values. "
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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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"execution_count": 68,
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"id": "5a134433",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"text/plain": [
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"solve_upper_triangular! (generic function with 1 method)"
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]
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},
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"execution_count": 68,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"function solve_upper_triangular(U,y)\n",
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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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@@ -108,18 +125,34 @@
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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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"execution_count": 69,
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"id": "b92332f7",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5-element Vector{Float64}:\n",
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" 1.0000000000000009\n",
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" 1.999999999999999\n",
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" 2.9999999999999964\n",
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" 4.000000000000002\n",
|
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" 5.000000000000005"
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]
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},
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"execution_count": 69,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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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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"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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"id": "962ec4a9",
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"metadata": {},
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"source": [
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"We can test if the obtained solution is correct using `@test`:"
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@@ -127,19 +160,30 @@
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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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"execution_count": 70,
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"id": "0e336c85",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
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]
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},
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"execution_count": 70,
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"metadata": {},
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||||
"output_type": "execute_result"
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}
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],
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"source": [
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"using Test\n",
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"@test sol ≈ x"
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"@test sol ≈ [1.0; 2.0; 3.0; 4.0; 5.0]"
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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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"id": "28dff449",
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||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": []
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||||
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@@ -18,4 +18,4 @@
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||||
var myIframe = document.getElementById("notebook");
|
||||
iFrameResize({log:true}, myIframe);
|
||||
});
|
||||
</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="Thursday 24 August 2023 14:59">Thursday 24 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
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||||
@@ -18,4 +18,4 @@
|
||||
var myIframe = document.getElementById("notebook");
|
||||
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="Thursday 24 August 2023 14:59">Thursday 24 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
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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="Thursday 24 August 2023 14:59">Thursday 24 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
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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="Thursday 24 August 2023 14:59">Thursday 24 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</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="Thursday 24 August 2023 14:59">Thursday 24 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</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="../../getting_started_with_julia/">« Getting started</a><a class="docs-footer-nextpage" href="../julia_async/">Tasks and 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="Thursday 24 August 2023 14:59">Thursday 24 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="../../getting_started_with_julia/">« Getting started</a><a class="docs-footer-nextpage" href="../julia_async/">Tasks and 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="Friday 25 August 2023 13:20">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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||||
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||||
@@ -18,4 +18,4 @@
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Reference in New Issue
Block a user