{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### TD L3 Algèbre et géométrie - Feuille 2 ex. 3 avec Sagemath\n", "\n", "*F-X. Dehon - 3 oct. 2020 - dehon[@]unice.fr*\n", "\n", "Enoncé :
\n", "\n", " \n", "\n", "\n", "\n", "\n", "
\n", "Les éléments de $E=\\mathbb{R}_2[X]$ dans l'énoncé sont ici déclarés comme des fonctions de la variable x sans précision sur son type.\n" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a=1/sqrt(2);\n", "print a.parent(),\",\",a.category()\n", "print a,AA(a),RR(a)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Plusieurs façons de déclarer une fonction, pas strictement équivalentes :" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a=1/sqrt(2)\n", "P0(x)=x+a\n", "Q0=lambda x:x+a\n", "def R0(x):return(x+a)\n", "print P0,\",\",Q0,\",\",R0\n", "print P0==Q0,bool(P0(x)==Q0(x))\n", "print P0==R0,bool(P0(x)==R0(x))\n", "\n", "print P0.derivative()\n", "try:print Q0.derivative()\n", "except Exception as e: print \"Q0.derivative() :\",e\n", "print Q0(x).derivative(x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "##### Définition de la forme $\\langle-,-\\rangle$ nommée $B$ ci-dessous, de l'application $u$, coordonnées et matrices" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def B(P,Q):return(integrate(P(x)*Q(x),x,-1,1))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "print B(lambda x:x, lambda x:x^3)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "A=matrix(3,3,lambda i,j:B(lambda x:x^i,lambda x:x^j));pretty_print(A)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "var('a b c')\n", "X=matrix([a,b,c]).transpose();pretty_print(X)\n", "print (X.transpose()*A*X)[0,0].expand()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "P(x)=a+b*x+c*x^2\n", "print B(P,P).expand()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def u(P):return((1-x^2)*(P.derivative(x))+(2*x+1)*P(x))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "print u(P)(x).expand()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "print B(u(P),u(P)).expand()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def coef(P):return([P(0),P.derivative(x)(0),P.derivative(x,2)(0)/2])" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "print coef(P)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "P1(x)=1;P2(x)=x;P3(x)=x^2\n", "Base=[P1,P2,P3]\n", "Mu=matrix([coef(u(e)) for e in Base]).transpose();pretty_print(Mu)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "print (X.transpose()*Mu.transpose()*A*Mu*X)[0,0].expand()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "##### Orthonormalisation de Gram-Schmidt de $(P1,P2,P3)$, coordonnées dans la nouvelle base" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def N(P):return(P/sqrt(B(P,P)))\n", "def proj(P,base):return(sum(B(P,L)*L for L in base))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L1=N(P1);print L1(x)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L2p=P2-proj(P2,[L1])\n", "L2=N(P2);print L2(x)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L3p=P3-proj(P3,[L1,L2])\n", "L3=N(L3p);print L3(x)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def coefL(P):return([B(P,L) for L in [L1,L2,L3]])" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "MLu=matrix([coefL(u(L)) for L in [L1,L2,L3]]).transpose()\n", "pretty_print(MLu)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "baseL=[L1,L2,L3]\n", "pretty_print(matrix(3,3,lambda i,j:B(baseL[i],baseL[j])))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "P=a*L1+b*L2+c*L3\n", "print B(u(P),u(P)).expand()" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "X=matrix([a,b,c]).transpose()\n", "print (X.transpose()*MLu.transpose()*MLu*X)[0,0].expand()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "##### Autre définition de $E$ dans Sagemath" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ ], "source": [ "print sqrt(2).parent()\n", "print SR,\",\",AA" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "S.=SR[] #AA[]\n", "print 1.parent(),\",\",S(1).parent()\n", "print 1==S(1)\n", "try:print 1.derivative()\n", "except Exception as e:print(e)\n", "print S(1).derivative()\n", "P=1+X+X^2;print integrate(P(x),x,-1,1)\n", "print B(P,P)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def u(P):return((1-X^2)*(P.derivative())+(2*X+1)*P)\n", "print u(X^2)\n", "v(P)=(1-X^2)*(P.derivative())+(2*X+1)*P\n", "print v(X^2),\"bizarre !!\"\n", "w(P)=derivative(P);print w(X^3);print derivative(X^3)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "##### Scholies" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "reset('P Q R')\n", "try:print P.parent()\n", "except Exception as e: print(e)" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath (system-wide)", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": -1, "url": "https://www.sagemath.org/" } }, "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 4 }