...

10_Poisson1d_function.ipynb

+{
+ "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
+}