{ "metadata": { "name": "", "signature": "sha256:952084081dee0b0cd795e77642b0eafd6ba1b4521246f9779ca409aad55a1b07" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Aproxima\u00e7\u00e3o de fun\u00e7\u00f5es pelo m\u00e9todo dos m\u00ednimos quadrados\n", "\n", "Vamos substituir a tabela $T$ por uma fun\u00e7\u00e3o $f:[a,b]\\to \\mathbb{R}$ e tomar uma fam\u00edlia de fun\u00e7\u00f5es $\\mathbf{F}=\\{ f_1, \\dots f_n\\}$, todas definidas no intervalo $[a,b]$. E considerar o problema de encontrar os coeficientes $a_1,\\dots,a_n$ que minimize o\n", "res\u00edduo quadr\u00e1tico $\\text{Res}(f,a_1,\\dots,a_n) = \\int_a^b(\\sum a_if_i(x) -f(x))^2 dx$.\n", "De modo parecido ao caso discreto resolvemos este problema ao solucionar o sistema normal associado:\n", "$$ Ax=b$$ \n", "onde as componentes da matriz de coeficientes s\u00e3o:\n", "$$ \\begin{gather} a_{ij} = \\langle f_i, f_j \\rangle = \\int_a^b f_i(x)f_j(x)dx\\\\\n", "b_i = \\langle f , f_i \\rangle = \\int_a^b f_i(x)f_j(x)dx \\end{gather} $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemplo\n", "Achar uma aproxima\u00e7\u00e3o de $f(x) = \\sin(x)$ por um polin\u00f4mio menor ou igual a tr\u00eas no intervalo $[-\\pi,\\pi]$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Uma boa oportunidade para usar o sympy!\n", "import sympy as sp\n", "from sympy.interactive import printing\n", "printing.init_printing()\n", "x, y, z = sp.symbols(\"x y z\")\n", "f, g, h = map(sp.Function, 'fgh')\n", "f1 = map(sp.Function, 'f1')\n", "f2 = map(sp.Function, 'f2')\n", "f3 = map(sp.Function, 'f3')\n", "f4 = map(sp.Function, 'f4')\n", "f = sp.sin(x)\n", "f1 = 1\n", "f2 = x\n", "f3= x**2\n", "f4= x**3\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "* C\u00e1lculo do sistema normal" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def escalar(f,g):\n", " ''' produto de duas fun\u00e7\u00f5es entre -Pi e Pi '''\n", " return sp.integrate(f*g,(x,-sp.pi,sp.pi))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "sp.integrate(f1*f3,(x,0,1))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{1}{3}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"metadata": {}, "outputs": [], "prompt_number": 18 }, { "cell_type": "code", "collapsed": false, "input": [ "A1 = [escalar(f1,g) for g in F]\n", "A2 = [escalar(f2,g) for g in F]\n", "A3 = [escalar(f3,g) for g in F]\n", "A4 = [escalar(f4,g) for g in F]\n", "A = sp.Matrix([A1,A2,A3,A4])\n", "A" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}2 \\pi & 0 & \\frac{2 \\pi^{3}}{3} & 0\\\\0 & \\frac{2 \\pi^{3}}{3} & 0 & \\frac{2 \\pi^{5}}{5}\\\\\\frac{2 \\pi^{3}}{3} & 0 & \\frac{2 \\pi^{5}}{5} & 0\\\\0 & \\frac{2 \\pi^{5}}{5} & 0 & \\frac{2 \\pi^{7}}{7}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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\\pi^{3}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ "\u23a1 0 \u23a4\n", "\u23a2 \u23a5\n", "\u23a2 2\u22c5\u03c0 \u23a5\n", "\u23a2 \u23a5\n", "\u23a2 0 \u23a5\n", "\u23a2 \u23a5\n", "\u23a2 3\u23a5\n", "\u23a3-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a6" ] } ], "prompt_number": 20 }, { "cell_type": "code", "collapsed": false, "input": [ "a = A**(-1)*b\n", "a" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}0\\\\- \\frac{1}{8 \\pi^{5}} \\left(- 1260 \\pi + 210 \\pi^{3}\\right) + \\frac{75}{4 \\pi^{2}}\\\\0\\\\- \\frac{105}{4 \\pi^{4}} + \\frac{1}{8 \\pi^{7}} \\left(- 2100 \\pi + 350 \\pi^{3}\\right)\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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sp.lambdify(x,g,\"numpy\")\n", "hl = sp.lambdify(x,h, \"numpy\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 33 }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 36 }, { "cell_type": "code", "collapsed": false, "input": [ "from matplotlib.pyplot import plot,grid" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 42 }, { "cell_type": "code", "collapsed": true, "input": [ "plot(t,fl(t),t,gl(t))\n", "grid()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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NTNdiXT/84k+wh/ysyZb51a4NuXOrE+WhuhXrxrpz63QdVy+FXggHMGQIVKsGpUu/W3b4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\nP4gWKbeRMqU9XTw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