From 2cfefa0ec9b798ce3bd3363d9275baa6c44d3565 Mon Sep 17 00:00:00 2001
From: "J. Gergaud" <gergaud@ap240726-01.local>
Date: Wed, 20 Nov 2024 11:23:55 +0100
Subject: [PATCH] foo
---
M2/TP-Newton.ipynb | 115 +++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 115 insertions(+)
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Automatic differentiation"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using LinearAlgebra\n",
+ "using ForwardDiff\n",
+ "A = [-1 2;1 4]; b=[1,1]; c=1\n",
+ "f(x) = 0.5*x'*A*x + b'*x +c\n",
+ "\n",
+ "analytic_∇f(x) = 0.5*(A' + A)*x+b\n",
+ "analytic_∇²f(x) = 0.5*(A' + A)\n",
+ "\n",
+ "∇f(x) = ForwardDiff.gradient(f, x)\n",
+ "∇²f(x) = ForwardDiff.hessian(f, x)\n",
+ "\n",
+ "x0 = [1,-1]\n",
+ "println(\"∇f(x0) = \", ∇f(x0))\n",
+ "println(\"analytic_∇f(x0) = \", analytic_∇f(x0))\n",
+ "println(\"analytic_∇f(x0) - ∇f(x0) = \", analytic_∇f(x0)- ∇f(x0))\n",
+ "\n",
+ "println(\"∇²f(x0) = \", ∇²f(x0))\n",
+ "println(\"analytic_∇²f(x0) = \", analytic_∇²f(x0))\n",
+ "println(\"analytic_∇²f(x0) - ∇²f(x0) = \", analytic_∇²f(x0)- ∇²f(x0))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Newton algorithm"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using LinearAlgebra\n",
+ "using ForwardDiff\n",
+ "\n",
+ "\"\"\"\n",
+ " Solve by Newton's algorithm the optimization problem Min f(x)\n",
+ " Case where f is a function from R^n to R\n",
+ "\"\"\"\n",
+ "\n",
+ "\n",
+ "function algo_Newton(f,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+ "# to complete\n",
+ " \n",
+ " # flag = 0 if the program stop on the first criteria\n",
+ " # flag = 2 if the program stop on the second criteria\n",
+ " # ...\n",
+ " return xₖ, flag, fₖ, ∇fₖ , k \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A = [1 0 ; 0 9/2]\n",
+ "b = [0,0]; c=0.\n",
+ "f1(x) = 0.5*x'*A*x + b'*x +c\n",
+ "x0 = [1000,-20]\n",
+ "println(\"Results for Newton on f1 : \", algo_Newton(f1,x0))\n",
+ "println(\"eigen value of 0.5(A^T+A) = \", 0.5*eigen(A'+A).values)\n",
+ "using Plots;\n",
+ "x=range(-10,stop=10,length=100)\n",
+ "y=range(-10,stop=10,length=100)\n",
+ "f11(x,y) = f1([x,y])\n",
+ "p1 = plot(x,y,f11,st=:contourf)\n",
+ "\n",
+ "A = [-1 0 ; 0 3]\n",
+ "f2(x) = 0.5*x'*A*x + b'*x +c\n",
+ "println(\"Results for Newton on f2 : \", algo_Newton(f2,x0))\n",
+ "println(\"eigen value of 0.5(A^T+A) = \", 0.5*eigen(A'+A).values)\n",
+ "f21(x,y) = f1([x,y])\n",
+ "p2 = plot(x,y,f21,st=:contourf)\n",
+ "# Rosenbrock function\n",
+ "x=range(-1.5,stop=2,length=100)\n",
+ "y=range(-0.5,stop=3.,length=100)\n",
+ "f3(x) = (1-x[1])^2 + 100*(x[2]-x[1]^2)^2\n",
+ "f31(x,y) = f3([x,y])\n",
+ "p3 = plot(x,y,f31,st=:contourf)\n",
+ "x0 = [1.1,1.2]\n",
+ "\n",
+ "println(\"Results for Newton on f3 : \", algo_Newton(f3,[1.1,1.2]))\n",
+ "\n",
+ "println(\"Results for Newton on f3 : \", algo_Newton(f3,[3.,0.]))\n",
+ "plot(p1,p2,p3)"
+ ]
+ }
+ ],
+ "metadata": {
+ "language_info": {
+ "name": "python"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
--
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