From e599ffd68404544ba620b5ba55a2705897d44ffd Mon Sep 17 00:00:00 2001
From: Frederic Weidling <fweidli@client61.num.math.uni-goettingen.de>
Date: Fri, 9 Dec 2016 12:28:49 +0100
Subject: [PATCH] blatt7
---
07_Newtonverfahren.ipynb | 227 +++++++++++++++++++++++++++++++++++++++
1 file changed, 227 insertions(+)
create mode 100644 07_Newtonverfahren.ipynb
diff --git a/07_Newtonverfahren.ipynb b/07_Newtonverfahren.ipynb
new file mode 100644
index 0000000..41c3e81
--- /dev/null
+++ b/07_Newtonverfahren.ipynb
@@ -0,0 +1,227 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Aufgabe 27\n",
+ "## Teil 1\n",
+ "Schreiben Sie ein allgemeines Newtonverfahren, dass Funktion und Ableitung übergeben bekommt und testen Sie an $f(x)=x^2-4$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Nullstelle: 2.000e+00, Funktionswert 0.000e+00, Iterationen 8\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy.linalg import norm, solve\n",
+ "\n",
+ "def newton(func,jacobian,x0,tol):\n",
+ " k=0\n",
+ " delta_x=solve(jacobian(x0),-func(x0))\n",
+ " x=x0+delta_x\n",
+ " \n",
+ " while True:\n",
+ " k=k+1\n",
+ " delta_x_new=solve(jacobian(x),-func(x))\n",
+ " x=x+delta_x_new\n",
+ " q=norm(delta_x_new)/norm(delta_x)\n",
+ " if q>=1:\n",
+ " print('Verfahren scheint nicht zu konvergieren')\n",
+ " return x,k\n",
+ " if q/(1-q)*norm(delta_x_new)<=tol:\n",
+ " return x,k\n",
+ " else:\n",
+ " delta_x=delta_x_new.copy()\n",
+ "\n",
+ "from numpy import array \n",
+ " \n",
+ "f=lambda t: array([t[0]**2-4])\n",
+ "df=lambda t: array([[2*t[0]]])\n",
+ "\n",
+ "x,k=newton(f,df,array([50.]),1e-8)\n",
+ "print('Nullstelle: {0:1.3e}, Funktionswert {1:1.3e}, Iterationen {2}'.format(float(x),float(f(x)),k) ) #"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": false
+ },
+ "source": [
+ "## Teil 2\n",
+ "Nun für die zweite gegebene Funktion"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "from numpy import zeros,eye\n",
+ "\n",
+ "def f_matrix_eval(x,A):\n",
+ " y=zeros(x.shape)\n",
+ " n=max(x.shape)\n",
+ " y[0:n-1]=A@x[0:n-1]-x[n-1]*x[0:n-1]\n",
+ " y[n-1]=x[0:n-1]@x[0:n-1]-1\n",
+ " return y\n",
+ "\n",
+ "def f_matrix_jacobi(x,A):\n",
+ " n=max(x.shape)\n",
+ " J=zeros([n,n])\n",
+ " J[0:n-1,0:n-1]=A-x[n-1]*eye(n-1)\n",
+ " J[0:n-1,n-1]=-x[0:n-1]\n",
+ " J[n-1,0:n-1]=2*x[0:n-1]\n",
+ " return J\n",
+ "\n",
+ "def f_poisson_eval(x):\n",
+ " P=array([[2.0,-1,0],[-1,2,-1],[0,-1,2]])\n",
+ " f=f_matrix_eval(x,P)\n",
+ " return f\n",
+ "\n",
+ "def f_poisson_jacobi(x):\n",
+ " P=array([[2.0,-1,0],[-1,2,-1],[0,-1,2]])\n",
+ " J=f_matrix_jacobi(x,P)\n",
+ " return J"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finden der ersten Nullstelle bei $x=(\\frac{1}{2},\\frac{1}{\\sqrt{2}},\\frac{1}{2},2-\\sqrt{2})$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Nullstelle: [ 0.5 0.70710678 0.5 0.58578644],\n",
+ "Abstand: 1.0114613123884174e-15,\n",
+ "Funktionswert: [ 3.88578059e-16 6.66133815e-16 3.88578059e-16 1.33226763e-15], \n",
+ "Iterationen: 4\n"
+ ]
+ }
+ ],
+ "source": [
+ "x,k=newton(f_poisson_eval,f_poisson_jacobi,array([1.,1.0,0.,0]),1e-8)\n",
+ "xtrue=array([0.5,2**(-1./2),0.5,2-2**(1./2)])\n",
+ "print('Nullstelle: {0},\\nAbstand: {1},\\nFunktionswert: {2}, \\nIterationen: {3}'.format(x,norm(x-xtrue),f_poisson_eval(x),k) ) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finden der zweiten Nullstelle bei $x=(\\frac{1}{\\sqrt{2}},0,-\\frac{1}{\\sqrt{2}},2)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Nullstelle: [ 0.70710678 0. -0.70710678 2. ],\n",
+ "Abstand: 1.1276404038266872e-12,\n",
+ "Funktionswert: [ 0.00000000e+00 0.00000000e+00 0.00000000e+00 2.25530705e-12], \n",
+ "Iterationen: 3\n"
+ ]
+ }
+ ],
+ "source": [
+ "x,k=newton(f_poisson_eval,f_poisson_jacobi,array([1.0,0,-1.,2.]),1e-8)\n",
+ "xtrue=array([2**(-1./2),0,-2**(-1./2),2.])\n",
+ "print('Nullstelle: {0},\\nAbstand: {1},\\nFunktionswert: {2}, \\nIterationen: {3}'.format(x,norm(x-xtrue),f_poisson_eval(x),k) ) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finden der dritten Nullstelle bei $x=(\\frac{1}{2},-\\frac{1}{\\sqrt{2}},\\frac{1}{2},2+\\sqrt{2})$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Nullstelle: [ 0.5 -0.70710678 0.5 3.41421356],\n",
+ "Abstand: 0.0,\n",
+ "Funktionswert: [ 0. 0. 0. 0.], \n",
+ "Iterationen: 4\n"
+ ]
+ }
+ ],
+ "source": [
+ "x,k=newton(f_poisson_eval,f_poisson_jacobi,array([1.,-1,1,3.]),1e-8)\n",
+ "xtrue=array([1./2,-2**(-1./2),1./2,2+2**(1./2)])\n",
+ "print('Nullstelle: {0},\\nAbstand: {1},\\nFunktionswert: {2}, \\nIterationen: {3}'.format(x,norm(x-xtrue),f_poisson_eval(x),k) ) "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.5.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
--
GitLab