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Merge branch 'main' of github.com:fverdugo/XM_40017 into main

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Francesc Verdugo 2023-08-25 15:27:54 +02:00
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notebooks/LEQ.ipynb
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@ -2,53 +2,59 @@
"cells": [ "cells": [
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "599913a3", "id": "9c32b051",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# Solving Linear Equations\n", "# Solving Linear Equations\n",
"\n", "\n",
"## Serial Algorithm\n", "## Serial Algorithm\n",
"First, we construct a linear equations system $Ax=b$." "To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 66,
"id": "fe3b5b02", "id": "2b369b73",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
"n = 5\n", "A = [1.0 4.0 5.0 8.0 1.0; \n",
"A = rand(-10.0:10.0, (n, n))" " 2.0 -1.0 4.0 3.0 0.0; \n",
] " 7.0 6.0 3.0 -4.0 5.0; \n",
}, " -3.0 4.0 2.0 2.0 2.0; \n",
{ " 0.0 -4.0 2.0 1.0 2.0]\n",
"cell_type": "code", "\n",
"execution_count": null, "b = [61.0; 24.0; 37.0; 29.0; 12.0];"
"id": "57c912fb",
"metadata": {},
"outputs": [],
"source": [
"x = rand(-10.0:10.0, n)\n",
"b = A * x"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "4baa0681", "id": "53124eb8",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$. " "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!"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 67,
"id": "11d255e0", "id": "7a7b926a",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"convert_to_upper_triangular! (generic function with 1 method)"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"function convert_to_upper_triangular(A,b)\n", "function convert_to_upper_triangular!(A,b)\n",
" n = size(A,1)\n",
" # Upper Triangularization: convert Ax=b to Ux=y\n", " # Upper Triangularization: convert Ax=b to Ux=y\n",
" for k in 1:n\n", " for k in 1:n\n",
" for j in k+1:n\n", " for j in k+1:n\n",
@ -72,20 +78,31 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "02c47593", "id": "78ef1849",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The function in the following cell solves the upper triangular equation system using backwards substitution. " "The function in the following cell solves the upper triangular equation system using backwards substitution. Note that the function alters the input values. "
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 68,
"id": "3c19497c", "id": "5a134433",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"solve_upper_triangular! (generic function with 1 method)"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"function solve_upper_triangular(U,y)\n", "function solve_upper_triangular!(U,y)\n",
" n = size(U,1)\n", " n = size(U,1)\n",
" for step in reverse(1:n)\n", " for step in reverse(1:n)\n",
" if U[step,step] == 0\n", " if U[step,step] == 0\n",
@ -108,18 +125,34 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 69,
"id": "2c24d85e", "id": "b92332f7",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"5-element Vector{Float64}:\n",
" 1.0000000000000009\n",
" 1.999999999999999\n",
" 2.9999999999999964\n",
" 4.000000000000002\n",
" 5.000000000000005"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"U,y = convert_to_upper_triangular(A,b)\n", "U,y = convert_to_upper_triangular!(A,b)\n",
"sol = solve_upper_triangular(U,y)" "sol = solve_upper_triangular!(U,y)"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "1c356b5a", "id": "962ec4a9",
"metadata": {}, "metadata": {},
"source": [ "source": [
"We can test if the obtained solution is correct using `@test`:" "We can test if the obtained solution is correct using `@test`:"
@ -127,19 +160,30 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 70,
"id": "5c71828d", "id": "0e336c85",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"using Test\n", "using Test\n",
"@test sol ≈ x" "@test sol ≈ [1.0; 2.0; 3.0; 4.0; 5.0]"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": null,
"id": "fe3c5374", "id": "28dff449",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [] "source": []