{
  "metadata": {
    "name": "",
    "signature": "sha256:a4126d6c54dc8683b5daa028f38743ed277fdf4bf63f2b5b535cc1b37c666f55"
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  "nbformat": 3,
  "nbformat_minor": 0,
  "worksheets": [
    {
      "cells": [
        {
          "cell_type": "markdown",
          "metadata": {},
          "source": [
            "# Vandermonde Conditioning"
          ]
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "import numpy as np\n",
            "import numpy.linalg as la\n",
            "import scipy.special as sp\n",
            "import matplotlib.pyplot as pt"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [],
          "prompt_number": 20
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "n = 5\n",
            "\n",
            "nodes = np.linspace(-1, 1, n)\n",
            "#nodes = sp.legendre(n).weights[:, 0]\n",
            "\n",
            "pt.plot(nodes, 0*nodes, \"o\")"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [
            {
              "metadata": {},
              "output_type": "pyout",
              "prompt_number": 21,
              "text": [
                "[<matplotlib.lines.Line2D at 0x7f6ea415b438>]"
              ]
            },
            {
              "metadata": {},
              "output_type": "display_data",
              "png": 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              "text": [
                "<matplotlib.figure.Figure at 0x7f6ea425a080>"
              ]
            }
          ],
          "prompt_number": 21
        },
        {
          "cell_type": "heading",
          "level": 3,
          "metadata": {},
          "source": [
            "Monomials"
          ]
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "exponents = np.arange(n)\n",
            "exponents"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [
            {
              "metadata": {},
              "output_type": "pyout",
              "prompt_number": 22,
              "text": [
                "array([0, 1, 2, 3, 4])"
              ]
            }
          ],
          "prompt_number": 22
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "exponents[:, np.newaxis]"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [
            {
              "metadata": {},
              "output_type": "pyout",
              "prompt_number": 23,
              "text": [
                "array([[0],\n",
                "       [1],\n",
                "       [2],\n",
                "       [3],\n",
                "       [4]])"
              ]
            }
          ],
          "prompt_number": 23
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "vdm = nodes ** exponents[:, np. newaxis]\n",
            "\n",
            "if n < 5:\n",
            "    print(vdm)"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [],
          "prompt_number": 24
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "print(la.cond(vdm))"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [
            {
              "output_type": "stream",
              "stream": "stdout",
              "text": [
                "4.30037521416\n"
              ]
            }
          ],
          "prompt_number": 29
        },
        {
          "cell_type": "heading",
          "level": 3,
          "metadata": {},
          "source": [
            "Legendre polynomials"
          ]
        },
        {
          "cell_type": "code",
          "collapsed": false,
          "input": [
            "vdm = np.array([sp.eval_legendre(i, nodes) for i in range(n)]).T\n",
            "\n",
            "print(la.cond(vdm))"
          ],
          "language": "python",
          "metadata": {},
          "outputs": [
            {
              "output_type": "stream",
              "stream": "stdout",
              "text": [
                "4.30037521416\n"
              ]
            }
          ],
          "prompt_number": 30
        }
      ],
      "metadata": {}
    }
  ]
}