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-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Résolution de systèmes linéaires "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "D'abord quelques exemples d'utilisations de la factorisation LU et de Choleski, ensuite on passe aux methodes itéreatives\n",
-    "\n",
-    "## Méthodes Directes"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercice \n",
-    "\n",
-    "On considère le système linéaire $A\\mathbf{x} = \\mathbf{b}$ où :\n",
-    "\n",
-    "$$A = \\left(\n",
-    "  \\begin{array}{ccc}\n",
-    "          3 & 6 & 7 \\\\[0.5em]\n",
-    "          1 & 1 & 4 \\\\[0.5em]\n",
-    "          2 & 4 & 8\n",
-    "        \\end{array} \\right), \\quad\n",
-    "  \\mathbf{b} = \\left( \\begin{array}{c} 4 \\\\[0.5em] 5 \\\\[0.5em] 6 \\end{array} \n",
-    "\\right) \\, .$$\n",
-    "      \n",
-    "1. Calculer la factorisation $LU$ de la matrice $A$ avec Python:\n",
-    "\n",
-    "2. Résoudre le système linéaire $A \\mathbf{x} = \\mathbf{b}$ en utilisant la factorisation trouvée au point\n",
-    "précédent (Ne plus utiliser Python. Mais ici on va le faire)\n",
-    "\n",
-    "3. Calculer le déterminant de la matrice $A$ en utilisant sa factorisation $LU$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# importing libraries used in this book\n",
-    "import numpy as np\n",
-    "import scipy.linalg as linalg\n",
-    "import pprint"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "A = np.array([[3, 6, 7],\n",
-    "              [1, 1, 4],\n",
-    "              [2, 4, 8] ])\n",
-    "\n",
-    "# LU factorisation with pivoting\n",
-    "P, L, U = linalg.lu(A)\n",
-    "\n",
-    "print(\"A = P L U\")\n",
-    "pprint.pprint(P.dot(L.dot(U)) )\n",
-    "\n",
-    "print(\"P:\")\n",
-    "pprint.pprint(P)\n",
-    "\n",
-    "print (\"L:\")\n",
-    "pprint.pprint(L)\n",
-    "\n",
-    "print (\"U:\")\n",
-    "pprint.pprint(U)\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 2\n",
-    "\n",
-    "Si $P$ est l'identité, $A = LU$.\n",
-    "\n",
-    "1. résoudre pour $\\mathbf y$ tel que $L \\mathbf y = \\mathbf{b}$ par substitution progressive\n",
-    "\n",
-    "2. résoudre pour $\\mathbf x$ tel que $U \\mathbf x = \\mathbf{y}$ par substitution retrograde\n",
-    "\n",
-    "Si $P$ n'est pas l'identité, $A = PLU$. \n",
-    "\n",
-    "Attention, une matrice de permutation est une matrice orthogonale car les colonnes sont orthonormée.\n",
-    "Donc $P^{-1} = P^T$. Du coup : $P^TA = LU$ et \n",
-    "$$A\\mathbf x = \\mathbf b \\Leftrightarrow P^TA\\mathbf x = P^T \\mathbf b\\Leftrightarrow LU\\mathbf x = P^T \\mathbf b.$$\n",
-    "Alors il faut modifier les calculs précedants comme suit:\n",
-    "\n",
-    "1. résoudre pour $\\mathbf y$ tel que $L \\mathbf y = P^T\\mathbf{b}$ par substitution progressive\n",
-    "\n",
-    "2. résoudre pour $\\mathbf x$ tel que $U \\mathbf x = \\mathbf{y}$ par substitution retrograde\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "b = np.array([4,5, 6])\n",
-    "\n",
-    "y = np.linalg.solve(L,P.T.dot(b))\n",
-    "x = np.linalg.solve(U,y)\n",
-    "# Here useless since P is the identity\n",
-    "\n",
-    "print(x)\n",
-    "\n",
-    "# check the residual of the equation\n",
-    "print(b - A.dot(x))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 3\n",
-    "\n",
-    "$$\\det A = \\det P \\, \\det L \\, \\det U$$"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "linalg.det(P)*linalg.det(L)*linalg.det(U)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "*Les mineurs principaux** d'une matrice $A \\in \\mathbb{R}^{n \\times n}$ \n",
-    "sont les déterminants des matrices $A_p = (a_{i,j})_{1 \\leq i, j \\leq p}$, $p = 1, ..., n$.\n",
-    "\n",
-    "**Critère de Sylvester:** une matrice symétrique $A \\in \\mathbb{R}^{n\\times n}$ \n",
-    "est définie positive si et seulement si les mineurs\n",
-    "principaux de $A$ sont tous positifs."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercice \n",
-    "\n",
-    "On considère le système linéaire $A\\mathbf{x} = \\mathbf{b}$\n",
-    "où\n",
-    "\n",
-    "$$A = \\left(\n",
-    "  \\begin{array}{ccc}\n",
-    "  \\varepsilon & 1 & 2\\\\\n",
-    "  1 & 3 & 1 \\\\\n",
-    "  2 & 1 & 3 \\\\\n",
-    "  \\end{array}\n",
-    "  \\right).$$\n",
-    "  \n",
-    "1. Déterminer pour quelles valeurs du paramètre réel\n",
-    "$\\varepsilon \\in \\mathbb{R}$, la matrice $A$ est symétrique\n",
-    "définie positive \n",
-    "\n",
-    "2. Soit maintenant $\\varepsilon=0$. On veut résoudre le système\n",
-    "$A\\mathbf{x}=\\mathbf{b}$ par une méthode directe; quelle\n",
-    "factorisation de la matrice $A$ envisageriez-vous? Justifier\n",
-    "votre réponse.\n",
-    "\n",
-    "3. En considérant $\\varepsilon = 2$, vérifier que dans ce cas la\n",
-    "matrice $A$ est définie positive et calculer sa\n",
-    "factorisation de Cholesky $A = L L^T$.\n",
-    "\n",
-    "4. En supposant que $\\mathbf{b} = (1,1,1)^T$, résoudre le\n",
-    "système linéaire $A \\mathbf{x} = \\mathbf{b}$ en utilisant la\n",
-    "factorisation de Cholesky calculée au point c).\n",
-    "\n",
-    "Reference Python pour la factorisation de Cholesky :\n",
-    "[scipy.linalg.cholesky](https://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.linalg.cholesky.html)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 1 t 2\n",
-    "\n",
-    "cf corrigé"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 3"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "epsilon = 2\n",
-    "A = np.array([[epsilon, 1, 2],\n",
-    "              [1, 3, 1],\n",
-    "              [2, 1, 3] ])\n",
-    "\n",
-    "# Is A symmetric ?\n",
-    "print(f'max(abs(A-AT)) = {np.max(np.abs(A-A.T))} ')\n",
-    "\n",
-    "# What are the eigenvlues of $A$ ? (usually, use linalg.eig, but here A is symmetric, we can use eigh )\n",
-    "lk, v = linalg.eigh(A)\n",
-    "print(f'The eigenvalues of A are {lk}')\n",
-    "\n",
-    "# Cholesky factorisation: lower : return lower-triangular matrix, A = L^T L\n",
-    "L = linalg.cholesky(A, lower=True) \n",
-    "print(f\"\\n A = L^T L = {L.dot(L)}\\n\")\n",
-    "\n",
-    "print (f\"L = {L}\")\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Partie 4"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "b = np.array([1,1, 1])\n",
-    "\n",
-    "y = np.linalg.solve(L,b)\n",
-    "x = np.linalg.solve(L.T,y)\n",
-    "\n",
-    "print(x)\n",
-    "\n",
-    "# check the residual of the equation\n",
-    "print(b - A.dot(x))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Méthodes itératives"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exemple\n",
-    "\n",
-    "$$A=\\begin{pmatrix}\n",
-    "1 & 2 & 3 & 4 \\\\\n",
-    "5 & 6 & 7 & 8\\\\\n",
-    "9 & 10 & 11 & 12\\\\\n",
-    "13 & 14 & 15 & 16\n",
-    "\\end{pmatrix}.$$\n",
-    "\n",
-    "On a alors\n",
-    "$$D=\\begin{pmatrix}\n",
-    "1 & 0 & 0 & 0 \\\\\n",
-    "0 & 6 & 0 & 0\\\\\n",
-    "0 & 0 & 11 & 0\\\\\n",
-    "0 & 0 & 0 & 16\n",
-    "\\end{pmatrix};$$\n",
-    "la matrice d'itération de la méthode de Jacobi est\n",
-    "$$B_J = D ^{-1}(D - A) = I - D^{-1}A =\n",
-    "\\begin{pmatrix}\n",
-    "0 & -2 & -3 & -4 \\\\\n",
-    "-5/6 & 0 & -7/6 & -4/3\\\\\n",
-    "-9/11 & -10/11 & 0 & -12/11\\\\\n",
-    "-13/16 & -14/16 & -15/16 & 0\n",
-    "\\end{pmatrix}.$$"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "A = np.array([[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]])"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# Jacobi\n",
-    "# first diag extract the diagonal of A, the second builds a diagonal matrix starting from a vector\n",
-    "D = np.diag(np.diag(A)) \n",
-    "\n",
-    "Bj = np.linalg.solve(D,(D-A))\n",
-    "# What are the eigenvlues of $A$ ? \n",
-    "lk, v = linalg.eig(Bj)\n",
-    "print(f'The eigenvalues of Bj are {lk}\\n')\n",
-    "print(f'The spectral radius of Bj is {np.max(np.abs(lk)):.4f} > 1, therefore Jacobi does not converge')\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# Gauss-Seidel\n",
-    "# Return a copy of an array with elements above the k-th diagonal zeroed. \n",
-    "Pgs =  np.tril(A,k=0)\n",
-    "\n",
-    "Bgs = np.linalg.solve(Pgs,(Pgs-A))\n",
-    "# What are the eigenvlues of $A$ ? \n",
-    "lk, v = linalg.eig(Bgs)\n",
-    "print(f'The eigenvalues of Bgs are {lk}\\n')\n",
-    "print(f'The spectral radius of Bgs is {np.max(np.abs(lk)):.4f} > 1, therefore Gauss-Seidel does not converge')\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Bgs"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  }
- ],
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-  "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
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