From f4c7fd7429f8033f8016070943bbac9949d4b71b Mon Sep 17 00:00:00 2001
From: Frederic Weidling <fweidli@client61.num.math.uni-goettingen.de>
Date: Fri, 23 Dec 2016 09:04:19 +0100
Subject: [PATCH] ...
---
10_Poisson1d_function.ipynb | 168 ++++++++++++++++++++++++++++++++++++
1 file changed, 168 insertions(+)
create mode 100644 10_Poisson1d_function.ipynb
diff --git a/10_Poisson1d_function.ipynb b/10_Poisson1d_function.ipynb
new file mode 100644
index 0000000..40593fa
--- /dev/null
+++ b/10_Poisson1d_function.ipynb
@@ -0,0 +1,168 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Wirkung der 1d-Poissongleichung auf einen Vektor\n",
+ "Es soll eine Funktion erstellt werden, die die Wirkung der 1d-Poissongleichung auf einen Vektor x simuliert. Die Idee dafür ist es am Anfang und Ende von x eine zusätzliche 0 einzufügen (sogenanntes zero padding) und dann entsprechende Ausschnitte auf x zu addieren."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[0 1 2 3 0]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import pad, array\n",
+ "\n",
+ "a=array([1,2,3])\n",
+ "print(pad(a,1,mode='constant'))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Nun die Funktion, die die Wirkung der 1d-Poissonmatrix auf einen Vektor berechnet:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "def poisson1d(x):\n",
+ " h=pad(x,1,mode='constant')\n",
+ " y=2*x-h[:-2]-h[2:]\n",
+ " n=max(x.shape)+1\n",
+ " return y*n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Wir testen an zwei einfachen Beispielen, um uns von der Richtigkeit zu überzeugen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": false,
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[ 0 0 16]\n",
+ "[ 10. 0. 0. 0. 0. 0. 0. 0. 10.]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import ones\n",
+ "\n",
+ "print(poisson1d(a))\n",
+ "\n",
+ "b=ones(9)\n",
+ "print(poisson1d(b))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Als Hinweise für den 2d-Fall: "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "array([[0, 0, 0, 0],\n",
+ " [0, 1, 2, 0],\n",
+ " [0, 3, 4, 0],\n",
+ " [0, 0, 0, 0]])"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "c=array([[1,2],[3,4]])\n",
+ "pad(c,1, mode='constant')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[[1 2]\n",
+ " [3 4]]\n",
+ "[1 2 3 4]\n"
+ ]
+ }
+ ],
+ "source": [
+ "d=array([1,2,3,4])\n",
+ "n=max(d.shape)\n",
+ "e=d.reshape((int(n**(1/2)),int(n**(1/2))))\n",
+ "print(e)\n",
+ "print(e.ravel())"
+ ]
+ }
+ ],
+ "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
+}
--
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