{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Computing the weights in Simpson's rule"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import numpy.linalg as la"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We found the integrals:\n",
        "\n",
        "$$\n",
        "\\begin{align*}\n",
        "\\int_0^1 1 dx &= 1\\\\\n",
        "\\int_0^1 x dx &= \\frac 12 \\\\\n",
        "\\int_0^1 x^2 dx &= \\frac 13 \\\\\n",
        "\\end{align*}\n",
        "$$\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "integrals = np.array([1, 1/2, 1/3])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([ 0. ,  0.5,  1. ])"
            ]
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "nodes = np.linspace(0, 1, 3)\n",
        "nodes"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[ 1.  ,  0.  ,  0.  ],\n",
              "       [ 1.  ,  0.5 ,  0.25],\n",
              "       [ 1.  ,  1.  ,  1.  ]])"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "V = np.array([\n",
        "1+0*nodes,\n",
        "nodes,\n",
        "nodes**2\n",
        "]).T\n",
        "V"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now compute the weights:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([ 0.16666667,  0.66666667,  0.16666667])"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "la.inv(V).T.dot(integrals)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 0
}