diff --git a/M2/TP-introduction-NN.ipynb b/M2/TP-introduction-NN.ipynb
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index 0000000000000000000000000000000000000000..6ee9dd039518fa3376cf3678cb16de03b9dff284
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Neural Network\n",
+ "\n",
+ "Complete the descent with constant rate function"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "function my_descent(f,x0::Vector{<:Real};η=0.1,AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)# to complete\n",
+ "\n",
+ " return xₖ, flag, fₖ, ∇fₖ, k c\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "And apply this code for solving the following problem\n",
+ "\n",
+ "$$(P)\\left\\{\\begin{array}{l}\n",
+ "Min\\;f(\\beta) = \\frac{1}{n}\\|X\\beta - y\\|^2\\\\\n",
+ "\\beta\\in\\R^2\n",
+ "\\end{array}\\right.\n",
+ "$$\n",
+ "with\n",
+ "$$X = \\begin{pmatrix}\n",
+ "1 & x_1\\\\\n",
+ "\\vdots & \\vdots\\\\\n",
+ "1 & x_5\n",
+ "\\end{pmatrix}\n",
+ "\\;\\;\\textrm{and}\\;\\;\n",
+ "y = \\begin{pmatrix}\n",
+ "y_1\\\\\n",
+ "\\vdots\\\\\n",
+ "y_5\n",
+ "\\end{pmatrix}\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "You 'll print the first 7 iterations for the initual guess \n",
+ "$x0= (n/2)(9,1)$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "plaintext"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "using Plots\n",
+ "# n = 5\n",
+ "x1 = 1.; x2 = sqrt(5*9/2-x1^2);\n",
+ "x = [-x2,-x1,0.,x1,x2]\n",
+ "n = length(x)\n",
+ "a₁ = 1; a₀ = 2\n",
+ "y = a₁*x .+ a₀ # model\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "function descent_backtrac(f,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+ " # to complete\n",
+ "\n",
+ " return xₖ, flag, fₖ, ∇fₖ, k c\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "using Plots\n",
+ "x = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+ "y = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+ "f_contour(x,y) = f([x,y])\n",
+ "z = @. f_contour(x', y)\n",
+ "nb_levels = 7\n",
+ "x0 = (n/2)*[9,1]"
+ ]
+ }
+ ],
+ "metadata": {
+ "language_info": {
+ "name": "python"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/M2/TP-steepest-descent-quad.ipynb b/M2/TP-steepest-descent-quad.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..fb8a13ce5c7b6984490a2de8ec190abc1987ac5f
--- /dev/null
+++ b/M2/TP-steepest-descent-quad.ipynb
@@ -0,0 +1,158 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Steepest descent for a quadratic function\n",
+ "$f(x) = x^Ax + b^Tx + c$, where $A$ is symetric and positive-definite\n",
+ "\n",
+ "Complete the steepest_descent_quad function"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "function steepest_descent_quad(A::Matrix{<:Real},b::Vector{<:Real},c::Real,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+ "# to complete\n",
+ "\n",
+ " return xₖ, flag, fₖ, ∇fₖ, k c\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "plaintext"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "using Plots\n",
+ "# n = 5\n",
+ "x1 = 1.; x2 = sqrt(5*9/2-x1^2);\n",
+ "x = [-x2,-x1,0.,x1,x2]\n",
+ "n = length(x)\n",
+ "X = [ones(n) x]\n",
+ "a₁ = 1; a₀ = 2\n",
+ "y = a₁*x .+ a₀ # model\n",
+ "A = (1/n)*X'*X # A=[1 0 ; 0 9]\n",
+ "println(\"A = \", A)\n",
+ "b = -(2/n)*X'*y\n",
+ "c = (1/n)*y'*y\n",
+ "println(\"b = \", b)\n",
+ "println(\"n= \",n)\n",
+ "\n",
+ "f(x) = x'*A*x + b'*x + c \n",
+ "xsol = -(A' + A)\\b\n",
+ "xx = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+ "yy = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+ "f_contour(x,y) = f([x,y])\n",
+ "z = @. f_contour(xx', yy)\n",
+ "nb_levels = 7\n",
+ "x0 = (n/2)*[9,1]\n",
+ "xₖ = x0\n",
+ "Xsol = zeros(nb_levels+1,2)\n",
+ "Xsol[1,:] = x0\n",
+ "p1 = plot()\n",
+ "for nbit in 1:nb_levels\n",
+ " xsol, flag, fsol, ∇f_xsol , nb_iter = steepest_descent_quad(A,b,c,x0,nbit_max=nbit)\n",
+ " plot!(p1,[xₖ[1],xsol[1]],[xₖ[2],xsol[2]],arrow=true)\n",
+ " xₖ = xsol\n",
+ " Xsol[nbit+1,:] = xsol\n",
+ "end\n",
+ "\n",
+ "levels = [f(Xsol[k,:]) for k in nb_levels+1:-1:1]\n",
+ "contour!(p1,xx,yy,z,levels=levels,cbar=false,color=:turbo)\n",
+ "\n",
+ "\n",
+ "xsol, flag, fsol, ∇f_xsol , nb_iter = algo_Newton(f,x0)\n",
+ "println()\n",
+ "println(\"Result with Newton's algorithm : \")\n",
+ "println(\"xsol = \", xsol)\n",
+ "println(\"flag = \", flag)\n",
+ "println(\"fsol = \", fsol)\n",
+ "println(\"∇f_xsol = \", ∇f_xsol)\n",
+ "println(\"nb_iter = \", nb_iter)\n",
+ "scatter!(p1,[xsol[1]],[xsol[2]])\n",
+ "plot!(p1,[x0[1],xsol[1]],[x0[2],xsol[2]],arrow=true)\n",
+ "plot(p1,legend=false)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "function descent_backtrac(f,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+ " # to complete\n",
+ "\n",
+ " return xₖ, flag, fₖ, ∇fₖ, k c\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "vscode": {
+ "languageId": "julia"
+ }
+ },
+ "outputs": [],
+ "source": [
+ "using Plots\n",
+ "x = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+ "y = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+ "f_contour(x,y) = f([x,y])\n",
+ "z = @. f_contour(x', y)\n",
+ "nb_levels = 7\n",
+ "x0 = (n/2)*[9,1]\n",
+ "xₖ = x0\n",
+ "p1 = plot()\n",
+ "for nbit in 1:nb_levels\n",
+ " xsol, flag, fsol, ∇f_xsol , nb_iter = descent_backtrac(f,x0,nbit_max=nbit)\n",
+ " #xsol, flag, fsol, ∇f_xsol , nb_iter = my_descent(f,x0,nbit_max=nbit)\n",
+ " println(\"xsol = \", xsol)\n",
+ " println(\"flag = \", flag)\n",
+ " println(\"fsol = \", fsol)\n",
+ " println(\"∇f_xsol = \", ∇f_xsol)\n",
+ " println(\"nb_iter = \", nb_iter)\n",
+ " plot!(p1,[xₖ[1],xsol[1]],[xₖ[2],xsol[2]],arrow=true)\n",
+ " xₖ = xsol\n",
+ "end\n",
+ "levels = [f(Xsol[k,:]) for k in nb_levels+1:-1:1]\n",
+ "contour!(p1,xx,yy,z,levels=levels,cbar=false,color=:turbo)\n",
+ "plot(p1,legend=false)"
+ ]
+ }
+ ],
+ "metadata": {
+ "language_info": {
+ "name": "python"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
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