{ "cells": [ { "cell_type": "markdown", "id": "5b03aa44", "metadata": {}, "source": [ "# Cálculo de autovalores\n", "#### https://meet.noysi.com/metodosnumericos1" ] }, { "cell_type": "markdown", "id": "45aa23c7-93ab-4aa9-9d99-034f3b5b622d", "metadata": {}, "source": [ "## Teorema de Gersgorin" ] }, { "cell_type": "markdown", "id": "8b0f9d74-f584-45e7-bbdc-1caf53f0ff41", "metadata": {}, "source": [ "La siguientes funciones calculan los discos del Teorema de Gersgorin y los representan." ] }, { "cell_type": "code", "execution_count": 1, "id": "a502845e-5ac5-4221-9f93-3f24dc372fb7", "metadata": {}, "outputs": [], "source": [ "def Gershgorin(A):\n", " return zip(A.diagonal(),vector([sum([abs(k) for k in fila]) for fila in A])-vector(map(abs,A.diagonal())) )\n", "def discosG(A):\n", " B=matrix(CDF,A)\n", " cr=Gershgorin(B)\n", " discos= sum([ circle([c.real(),c.imag()],r,fill=true,alpha=0.2) for c,r in cr])\n", " return discos" ] }, { "cell_type": "code", "execution_count": 2, "id": "85ac3292", "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "4 & -1 & 1 \\\\\n", "-1 & -3 & 1 \\\\\n", "1 & 2 & 5\n", "\\end{array}\\right) \\left[-3.426743094681910?, 3.757942566075653?, 5.668800528606257?\\right]$$" ], "text/plain": [ "[ 4 -1 1]\n", "[-1 -3 1]\n", "[ 1 2 5] [-3.426743094681910?, 3.757942566075653?, 5.668800528606257?]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = matrix([[4,-1,1],[-1,-3,1],[1,2,5]])\n", "show(A,A.eigenvalues())\n", "discosG(A)" ] }, { "cell_type": "code", "execution_count": null, "id": "f36b686f-e381-470e-bb94-d0554908757c", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "d712e005-e37e-4890-bf70-34fbe810b8f0", "metadata": {}, "source": [ "
\n", " Ejercicio 1. \n", "\n", "a) Aplicar el Teorema de Gersgorin para localizar los autovalores de la matriz \n", "$$A=\\left(\\begin{array}{rrr} 15.0 & -1.0 & 1.0 \\\\ 2.0 & -5.0 & 1.0 \\\\ 1.0 & 1.0 & -3.0 \\end{array}\\right). $$\n", "\n", "b) Aplicar el método de la potencia para aproximar el autovalor de módulo máximo. En cada paso, normaliza el vector. Detener el método cuando $\\|A v_k - \\lambda_k v_k\\|_2<10^{-2}$, donde $\\lambda_k$ es el autovalor aproximado obtenido y $v_k$ es el vector normalizado correspondiente a dicha iteración (que es una aproximación del autovalor). Aplicar el método con varios pasos y representar los valores obtenidos para $\\|A v_k - \\lambda_k v_k\\|_2<10^{-2}$. \n", " \n", "c) Aplicar el método de la potencia para aproximar el autovalor de módulo máximo, pero tomando como vector inicial uno de los autovectores correspondientes a un autovalor que no sea el de módulo máximo. Realizar $100$ iteraciones del método y representar el error residual $\\|A v_k - \\lambda_k v_k\\|_2$. \n", "\n", "d) Aplicar el método de la potencia inversa para aproximar el autovalor de módulo mínimo, con las mismas consideraciones que en el apartado b).\n", "\n", "e) Aplicar el método de la potencia inversa desplazada para encontrar el autovalor restante, con las mismas consideraciones que en el apartado b).\n", " \n", "
" ] }, { "cell_type": "code", "execution_count": 3, "id": "94fe814e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", "15 & -1 & 1 \\\\\n", "2 & -5 & 1 \\\\\n", "1 & 1 & -3\n", "\\end{array}\\right) \\left[-5.318425642342306?, -2.639723010334768?, 14.95814865267708?\\right]$$" ], "text/plain": [ "[15 -1 1]\n", "[ 2 -5 1]\n", "[ 1 1 -3] [-5.318425642342306?, -2.639723010334768?, 14.95814865267708?]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = matrix([[15,-1,1],[2,-5,1],[1,1,-3]])\n", "show(A,A.eigenvalues())\n", "discosG(A)" ] }, { "cell_type": "code", "execution_count": 43, "id": "ff0e1ba9", "metadata": {}, "outputs": [], "source": [ "v0 = vector(CDF,[1,1,1]) \n", "err = []" ] }, { "cell_type": "code", "execution_count": 51, "id": "f2d58a8e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "14.955471635604079 (0.9928793316010879, 0.10223756720332493, 0.06114010739050299) 0.007860072960139172\n" ] } ], "source": [ "while(true):\n", " v1 = A*v0\n", " l1 = v0.conjugate()*v1/(v0.conjugate()*v0)\n", " v1 = v1.normalized(2)\n", " err.append((A*v1-l1*v1).norm(2))\n", " if (A*v1-l1*v1).norm(2)< 10^-2:\n", " break\n", " v0 = copy(v1)\n", "print(l1,v1,(A*v1-l1*v1).norm(2))" ] }, { "cell_type": "code", "execution_count": 50, "id": "bc052ce5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "list_plot(err,plotjoined=true) .show(figsize=4)" ] }, { "cell_type": "code", "execution_count": 14, "id": "87df47ba", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(15.0, 0)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "for i in range(100):\n", " v1 = A*v0\n", " l1 = v0.conjugate()*v1/(v0.conjugate()*v0)\n", " v1 = v1.normalized(2)\n", " err.append((A*v1-l1*v1).norm(2))\n", " if (A*v1-l1*v1).norm(2)< 10^-2:\n", " break\n", " v0 = copy(v1)\n", "print(l1,v1,(A*v1-l1*v1).norm(2))" ] }, { "cell_type": "code", "execution_count": null, "id": "83997387", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "0549a4e4", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "c34612fc", "metadata": {}, "source": [ "
\n", " Ejercicio 2. \n", "\n", "a) Utilizar el método de la potencia (desplazando los autovalores si es necesario) para calcular el radio espectral de la matriz\n", "$$M=\\left(\\begin{array}{rrr} 2.0 & -2.0 & 1.0 \\\\ 3.0 & 2.0 & 1.0 \\\\ 1.0 & 1.0 & -1.0 \\end{array}\\right).$$\n", "Dibuja en cada paso del método el módulo del error residual $\\|A v_k - \\lambda_k v_k\\|$ obtenido. Detener el método cuando $\\|A v_k - \\lambda v_k\\|_2<10^{-2}$.\n", " \n", "b) Calcula el resto de los autovalores.\n", " \n", "
" ] }, { "cell_type": "code", "execution_count": null, "id": "b01a43cb", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "4c444e21", "metadata": {}, "source": [ "
\n", " Ejercicio 3. \n", " \n", "a) Aplicar 10 pasos del método QR para aproximar los autovalores de la matriz \n", "$$\\left(\\begin{array}{rrr} 15.0 & -1.0 & 1.0 \\\\ 2.0 & -5.0 & 1.0 \\\\ 1.0 & 1.0 & -3.0 \\end{array}\\right). $$\n", "La factorización QR de cada paso, calcularla usando el método `QR` de Sage.\n", "\n", "b) Obtener una factorización QR de la matriz anterior, utilizando reflexiones de Householder. \n", " \n", "
" ] }, { "cell_type": "code", "execution_count": null, "id": "8a8d48d9", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "fb7ddc10", "metadata": {}, "source": [ "
\n", " Ejercicio 4. \n", " \n", "Encontrar una matriz tal que uno de los discos de Gersgorin no contenga ningún autovalor.\n", " \n", "
" ] }, { "cell_type": "code", "execution_count": null, "id": "76231cf2", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.5", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.2" } }, "nbformat": 4, "nbformat_minor": 5 }