mirror of
https://github.com/fverdugo/XM_40017.git
synced 2025-12-30 02:38:31 +01:00
build based on 4a6041a
This commit is contained in:
Documenter.jl
207
dev/LEQ.ipynb
Normal file
207
dev/LEQ.ipynb
Normal file
@@ -0,0 +1,207 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9c32b051",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Solving Linear Equations\n",
|
||||
"\n",
|
||||
"## Serial Algorithm\n",
|
||||
"To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 66,
|
||||
"id": "2b369b73",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"A = [1.0 4.0 5.0 8.0 1.0; \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",
|
||||
"\n",
|
||||
"b = [61.0; 24.0; 37.0; 29.0; 12.0];"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "53124eb8",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"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",
|
||||
"execution_count": 67,
|
||||
"id": "7a7b926a",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"convert_to_upper_triangular! (generic function with 1 method)"
|
||||
]
|
||||
},
|
||||
"execution_count": 67,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"function convert_to_upper_triangular!(A,b)\n",
|
||||
" n = size(A,1)\n",
|
||||
" # Upper Triangularization: convert Ax=b to Ux=y\n",
|
||||
" for k in 1:n\n",
|
||||
" for j in k+1:n\n",
|
||||
" # Divide by pivot\n",
|
||||
" A[k,j] = A[k,j] / A[k,k]\n",
|
||||
" end\n",
|
||||
" b[k] = b[k] / A[k,k]\n",
|
||||
" A[k,k] = 1\n",
|
||||
" # Substract lower rows\n",
|
||||
" for i in k+1:n \n",
|
||||
" for j in k+1:n\n",
|
||||
" A[i,j]=A[i,j] - A[i,k] * A[k,j]\n",
|
||||
" end\n",
|
||||
" b[i] = b[i] - A[i,k] * b[k]\n",
|
||||
" A[i,k] = 0\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" return A, b #U,y\n",
|
||||
"end\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "78ef1849",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"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",
|
||||
"execution_count": 68,
|
||||
"id": "5a134433",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"solve_upper_triangular! (generic function with 1 method)"
|
||||
]
|
||||
},
|
||||
"execution_count": 68,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"function solve_upper_triangular!(U,y)\n",
|
||||
" n = size(U,1)\n",
|
||||
" for step in reverse(1:n)\n",
|
||||
" if U[step,step] == 0\n",
|
||||
" if y[step] != 0\n",
|
||||
" return \"No solution\"\n",
|
||||
" else\n",
|
||||
" return \"Infinity solutions\"\n",
|
||||
" end\n",
|
||||
" else\n",
|
||||
" # Backwards substitution\n",
|
||||
" y[step] = y[step] / U[step,step]\n",
|
||||
" end\n",
|
||||
" for row in reverse(1:step-1)\n",
|
||||
" y[row] -= U[row,step] * y[step]\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" return y \n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 69,
|
||||
"id": "b92332f7",
|
||||
"metadata": {},
|
||||
"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": [
|
||||
"U,y = convert_to_upper_triangular!(A,b)\n",
|
||||
"sol = solve_upper_triangular!(U,y)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "962ec4a9",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can test if the obtained solution is correct using `@test`:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 70,
|
||||
"id": "0e336c85",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
|
||||
]
|
||||
},
|
||||
"execution_count": 70,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"using Test\n",
|
||||
"@test sol ≈ [1.0; 2.0; 3.0; 4.0; 5.0]"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "28dff449",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Julia 1.9.1",
|
||||
"language": "julia",
|
||||
"name": "julia-1.9"
|
||||
},
|
||||
"language_info": {
|
||||
"file_extension": ".jl",
|
||||
"mimetype": "application/julia",
|
||||
"name": "julia",
|
||||
"version": "1.9.1"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Reference in New Issue
Block a user