{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as pt\n",
      "\n",
      "import scipy.sparse as sps"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We'll solve\n",
      "\n",
      "$u''+1000(1+x^2)u=0$ on $(-1,1)$\n",
      "\n",
      "with $u(-1)=3$ and $u(1)=-3$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 500\n",
      "mesh = np.linspace(-1, 1, n)\n",
      "h = mesh[1] - mesh[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 160
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = sps.diags(\n",
      "    [1,-2,1],\n",
      "    offsets=[-1,0,1], \n",
      "    shape=(n, n))\n",
      "\n",
      "if n < 10:\n",
      "    print A.todense()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 145
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Need to restrict to middle rows!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "second_deriv = ...\n",
      "\n",
      "if n < 10:\n",
      "    print second_deriv.todense()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 153
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "sps.diags(\n",
      "    [1,-2,1],\n",
      "    offsets=np.array([-1,0,1])+1,\n",
      "    shape=(n-2, n))/h**2\n",
      "-->\n",
      "(Edit this cell for solution)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "factor = sps.diags(\n",
      "    [1000*(1 + mesh[1:]**2)],\n",
      "    offsets=[1],\n",
      "    shape=(n-2, n))\n",
      "\n",
      "if n < 10:\n",
      "    print mesh[1:-1]\n",
      "    print\n",
      "    print factor.todense()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 154
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A_int = second_deriv+factor\n",
      "\n",
      "if n < 10:\n",
      "    print A_int.todense()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 155
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = sps.vstack([\n",
      "    sps.coo_matrix(([1], ([0],[0])), shape=(1, n)),\n",
      "    A_int,\n",
      "    sps.coo_matrix(([1], ([0],[n-1])), shape=(1, n)),\n",
      "    ])\n",
      "A = sps.csr_matrix(A)\n",
      "\n",
      "if n < 10:\n",
      "    print A.todense()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rhs = np.zeros(n)\n",
      "..."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 157
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "rhs[0] = 3\n",
      "rhs[-1] = -3\n",
      "-->\n",
      "(Edit this cell for solution.)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.sparse.linalg as sla\n",
      "\n",
      "sol = sla.spsolve(A, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 158
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pt.plot(mesh, sol)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 159,
       "text": [
        "[<matplotlib.lines.Line2D at 0x5d0e610>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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HB+JNLBUV9BzbgQFzbTC5Ykq7Uo8aS5ebEITUVr9E7bC+PkqghUngxU3KmiD1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\nSlWT+qRJ8go1KUqdg9TD9AO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       "text": [
        "<matplotlib.figure.Figure at 0x5b665d0>"
       ]
      }
     ],
     "prompt_number": 159
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 136
    }
   ],
   "metadata": {}
  }
 ]
}