{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. Feladatsor\n", "\n", "\n", "
\n", "### Spline interpoláció\n", "\n", "1.feladat. Írjuk fel azt az $S(x)$ másodfokú spline-t, amely illeszkedik az $(-1,2),\\ (0,1),\\ (2,-1)$ pontokra és $S'(2)=-2$.
\n", "\n", "2.feladat. Az $f(x)=1/x$ függvényt interpoláljuk másodfokú spline interpolációval az $x=1,\\ 2,\\ 3$ osztópontokon és $S'(1)=-1$.
\n", "\n", "3.feladat. Írjuk fel azt az $S(x)$ másodfokú spline interpolációt, amely illeszkedik a $(-1,2),\\ (0,1),\\ (2,-1), (3,-3)$ pontokra és $S(x)$ a $[2,3]$ intervallumon lineáris!
\n", "\n", "4.feladat.(Beadható) Megadható-e a $(-\\pi,-1),\\ (0,1),\\ (\\pi,-1)$ pontokon interpoláló másodfokú periodikus spline?
\n", "\n", "5.feladat. Tekintsük az $f(x)=\\sin\\Big(\\displaystyle\\frac{\\pi}{2}x\\Big)$ függvényt az $x=-1,\\ 0,\\ 1$ osztópontokon. Határozzuk meg az $f$-et interpoláló
\n", "\n", "(a) köbös természetes spline-t,
\n", "(b) köbös spline-t Hermite-féle peremfeltétellel.
\n", "\n", "6.feladat.(Beadható) Felírható-e olyan harmadfokú periodikus spline interpolációs polinom, amely illeszkedik a $(0,0),\\ (1,3),\\ (2,0)$ pontokra, ha azt az alábbi alakban keressük:
\n", "\n", "\\begin{equation}\n", "S(x)=\\begin{cases}\n", "S_1(x)=a_1x^3+b_1x^2+c_1x+d_1,\\ \\ \\ x\\in[0,1],&\\\\\n", "&\\\\\n", "S_2(x)=a_2(x-2)^3+b_2(x-2)^2+c_2(x-2)+d_2,\\ \\ \\ x\\in[1,2].&\n", "\\end{cases}\n", "\\end{equation}
\n", "\n", "7.feladat. Írjuk fel az $f$-et interpoláló Hermite-féle peremfeltételû harmadfokú spline-t, melyre $$f(-1)=1,\\ f'(-1)=-1,\\ f(0)=-1,\\ f(1)=1,\\ f'(1)=3.$$
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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4WEWgHoGf7LQOA0jaMjZispDVb94Dqbq6moj0er3bdYPBQEQVFRXOK5v+UmuK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\nghvJJ39jgwYYmOcRl0T0tn1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M3L17N/ttyInbd0txAontjuM5s5ldYX8BknA4HCtXriwp\nKXnggQckXA/b0dFx5cqVVatWeX6IXfzDH/4g9iSr733ve2o3QX1sZN7ZWXRiE8nOnDmjQpu4we27\npTiBlJycPGbMmOPHj3d2drruVVVUVEREc+fOVa9piurt7V25cuVHH3103333/fa3v1W7OSr4zW9+\n41YE7+3tZXM6tm3bRkS33XabOi1TyowZM0JCQjo6OlpbW113qfjss8+IKDIyUr2mKSckJISIuru7\n3a6zGR/so9Li9t1SnDGkwMDAhx56qKenZ/fu3c6L1dXVRUVFQUFBksy26uvrW7NmDUujnTt3Sjhy\nRkQLFy5MvZ5zDxX2UvidhPz8/JYsWUJEO3fudL3Obg1JjrVMTk4moqKiIrd+wFtvveX8qLS4fbcU\nanPVxsbGJ554oqWl5fHHHzebzfX19Xv37m1padm8eTO7P4VXWFj4wgsvENHcuXP9/f3dPrp69erp\n06er0S6VORyOu+66i2TaV7Szs/Pxxx+vra2dP3/+ww8/3Nvb+4c//KGioiIqKurdd9+VpH+QkZFR\nUlISHh7+k5/8JC4urqOj449//OOnn346YcKEwsJCUbfAP3Xq1J49e9if2eHaJpOJvVy+fLlzMSyf\n75ZCBRIRnTt37vnnn2dDc0Sk1+vXrVsnSRoR0TvvvDPAxgRvvPGGVJO/nSQMJCL6+uuvN2zY8MEH\nHzivzJ8/f/PmzZ6bS4mqq6vr1Vdfffvtt103tJ41a9bmzZunTp2qYsN86siRI+vWrev3Q7/+9a8f\nffRR50sO3y1FCyTGZrPV1dUFBQUlJCT4+YlTlgQYrvb2djZ0xAaW1G6OCvr6+srLy//5z3/6+fnN\nnDkzODhY7Rbxhat3SzEDCQAANAe9BwAA4AICCQAAuIBAAgAALiCQAACACwgkAADgAgIJAAC4gEAC\nAAAuIJAAAIALCCQAAOACAgkAALiAQAIAAC4gkAAAgAsIJAAA4AICCQAAuIBAAgAALiCQAACACwgk\nAADgwv8H6i7cnokLoVEAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "% Beepitett kobos spline \n", "x = [0 1 2.5 3.6 5 7 8.1 10];\n", "y = sin(x);\n", "xx = 0:.25:10;\n", "yy = spline(x,y,xx);\n", "plot(x,y,'o',xx,yy)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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opK4FZCkkObchKabTlBOQhUDiiP3Xcc4DCU0NcMfItb0BJgTbUsAUIZDgDnR7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\neTgqjpMMAumf/umfqqur9+3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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "% Beepitett kobos spline vegpontbeli nulla derivaltakkal\n", "x = -4:4;\n", "y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];\n", "cs = spline(x,[0 y 0]);\n", "xx = linspace(-4,4,101);\n", "plot(x,y,'o',xx,ppval(cs,xx),'-');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "### Legkisebb négyzetek elve és a Gauss-féle normálegyenlet\n", "\n", "8.feladat. Adjuk meg a legkisebb négyzetek elvén alapuló minimumfeladat megoldását.
\n", "\n", "9.feladat. Határozzuk meg a $(0,1),\\ (1,3),\\ (2,4),\\ (3,6)$ pontokat négyzetesen legjobban közelítõ egyenest!
\n", "\n", "10.feladat.(Beadható) Vizsgáljuk meg a Gauss-féle normál-egyenlet megoldhatóságát!
\n", "\n", "11.feladat. Tekintsünk egy periodikus folyamat egy alkalmas modelljét:\n", "$$F(t)=x_1+x_2\\cos(\\pi t)+x_3\\sin(\\pi t).$$\n", "A $0,\\ 1/2,\\ 2,\\ 5/2,\\ 4,\\ 9/2$ idpõpontok mérési eredményeibõl vizsgáljuk meg a feladat megoldhatóságát és ezzel egyetemben határozzuk meg a modell ismeretlen paramétereit.
\n", "\n", "12.feladat. Kétféleképpen határozzuk meg a $(2,1),\\ (4,3),\\ (6,2)$ pontokat négyzetesen legjobban közelítõ egyenest!
\n", "\n", "13.feldat. Írjuk fel a megadott $(t_i, f_i)$ pontokat négyzetesen legjobban közelítõ egyenest a Gauss-féle normál-egyenlet segítségével.\n", "\n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", " \n", "
t_i-2 | -1 | 0 | 1 | 2
f_i-4 | -2 | 1 | 2 | 4
\n", "\n", "Számítsuk ki a maradékvektor euklideszi hosszának négyzetét.
\n", "\n", "14.feladat. Írjuk fel a megadott $(t_i, f_i)$ pontokat négyzetesen legjobban közelítõ parabolát a Gauss-féle normál-egyenlet segítségével.\n", "\n", "\n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", " \n", "
t_i-2 | -1 | 1 | 2
f_i 3 | 1 | 0 | 2
\n", "\n", "Számítsuk ki a maradékvektor euklideszi hosszának négyzetét.
\n", "\n", "15.feladat. Írjuk fel a megadott $(t_i, f_i)$ pontokat négyzetesen legjobban közelítõ parabolát és harmadfokú polinomot a Gauss-féle normál-egyenlet segítségével.\n", "\n", "\n", "\n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", " \n", "
t_i-2 | 0 | 3
f_i1 | 3 | 4
" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A =\n", "\n", " 1 -2 4\n", " 1 -1 1\n", " 1 0 0\n", " 1 2 4\n", "\n", "\n", "ATranA =\n", "\n", " 4 -1 9\n", " -1 9 -1\n", " 9 -1 33\n", "\n", "\n", "ATranf =\n", "\n", " 7\n", " 1\n", " 13\n", "\n", "\n", "x =\n", "\n", " 2.4000\n", " 0.3500\n", " -0.2500\n", "\n", "\n", "Jnorma =\n", "\n", " 1.1000\n", "\n", "\n", "MatlabPolyfit =\n", "\n", " 2.4000\n", " 0.3500\n", " -0.2500\n" ] } ], "source": [ "\n", "[A,ATranA,ATranf,x,Jnorma,MatlabPolyfit]=legkisebbnegyzetek([-2 -1 0 2],[1 1 3 2],2)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "ans =\n", "\n", " -1/6 \n", " -3/10 \n", " 2/3\n" ] } ], "source": [ "rats(x)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "function [A,ATranA,ATranf,x,Jnorma,MatlabPolyfit]=legkisebbnegyzetek(t,f,n)\n", "%% A Gauss-féle normál-egyenlet megvalósítása n-edfokú polinomok esetén\n", "\n", "% A feladat: adott (t_i,f_i) alakú pontokra legkisebb négyzetek módszerének\n", "% megvalósítása Gauss-féle normál-egyenlettel\n", "%\n", "%\n", "\n", "\n", "%% Bemenõ paraméterek listája: \n", "\n", "% t alappontok\n", "% f t-hez tartozó értékek\n", "% n a közelítõ módszer rendje\n", "\n", "%% Elõkészületek \n", "\n", "e=ones(max(size(t)),1);\n", "if n==0\n", " A(:,1)=e;\n", " ATranA=(A'*A);\n", " ATranf=(A'*f');\n", " x=(A'*A)\\(A'*f');\n", " J=A*x-f';\n", " Jnorma=norm(J,2)^2;\n", "elseif n==1\n", " A(:,1)=e;\n", " A(:,2)=t';\n", " ATranA=(A'*A);\n", " ATranf=(A'*f');\n", " x=(A'*A)\\(A'*f');\n", " J=A*x-f';\n", " Jnorma=norm(J,2)^2;\n", "else n>=2\n", " A(:,1)=e;\n", " A(:,2)=t';\n", " for i=3:n+1\n", " A(:,i)=A(:,i-1).*t';\n", " end\n", " ATranA=(A'*A);\n", " ATranf=(A'*f');\n", " x=(A'*A)\\(A'*f');\n", " J=A*x-f';\n", " Jnorma=norm(J,2)^2;\n", "end\n", "\n", "%% Biztonsági összevetés a Matlab beépített függvényével\n", "\n", "MatlabPolyfit=rot90(polyfit(t,f,n)',2);\n", "\n" ] } ], "metadata": { "kernelspec": { "display_name": "Matlab", "language": "matlab", "name": "matlab" }, "language_info": { "codemirror_mode": "octave", "file_extension": ".m", "help_links": [ { "text": "MetaKernel Magics", "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" } ], "mimetype": "text/x-octave", "name": "matlab", "version": "0.11.0" } }, "nbformat": 4, "nbformat_minor": 0 }