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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Splines\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Piecewise continuous curves. $S:[a,b]\\to \\Re$\n",
      "\n",
      "$S = P_i (t) \\quad t_{i-1} \\leq t \\lt t_i$ for $i=0,...k$\n",
      "\n",
      "$P_i$ are polynomials. Chosen to gaurantee smoothness of $S$. $P_i^{(j)} (t_i) = P_{i+1}^{(j)} (t_i)$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The spline is defined by $\\{q_0(x) ... q_n(x)\\}$ for $n$ points $x_i$.\n",
      "\n",
      "For cubic splines have: \n",
      "\\begin{align}\n",
      "    q_i(x) &= a_i x^3 + b_i x^2 + c_i x + d_i \\\\\n",
      "    q_i(x_i) = y_i &= q_{i+1}(x_i) \\\\\n",
      "    &\\implies x_i^3 (a_{i} - a_{i+1}) + x_i^2 (b_{i} - b_{i+1}) + x_i (c_{i} - c_{i+1}) + (d_{i} - d_{i+1}) = 0 \\quad \\forall i\\\\\n",
      "    q'_i(x_i) &= q'_{i+1}(x_i) \\\\\n",
      "     &\\implies 3 x_i^2 (a_{i} - a_{i+1}) + 2 x_i (b_{i} - b_{i+1}) + (c_{i} - c_{i+1}) = 0 \\quad \\forall i\\\\\n",
      "     q''_i(x_i) &= q''_{i+1}(x_i) \\\\\n",
      "     &\\implies 6 x_i (a_{i} - a_{i+1}) + 2 (b_{i} - b_{i+1}) = 0 \\quad \\forall i\n",
      "\\end{align}\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Not useful:** Summing:\n",
      "\n",
      "\\begin{align}\n",
      "    0 &= a_0 (x_0^3 + 3 x_0^2 + 6 x_0) + b_0 (x_0^2 + 2x_0+2) + c_0 (x_0 + 1) + d_0 \\\\\n",
      "      &+ \\sum_{i=1}^{n} a_i \\left[ (x_i^3 - x_{i-1}^3) + 3(x_i^2 - x_{i-1}^2) + 6(x_i - x_{i-1})\\right]\n",
      "      + b_i\\left[(x_i^2 - x_{i-1}^2) + 2(x_i - x_{i-1})\\right] + c_i\\left[x_i - x_{i-1}\\right] \\\\\n",
      "\\end{align}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$q_i(x) = a_i x^3 + b_i x^2 + c_i x + d_i$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$q_i'(x) = 3 a_i x^2 + 2 b_i x + c_i$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$q_i''(x) = 6 a_i x + 2 b_i$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$x_i^3 a_i + x_i^2 b_i + x_i c_i + d_i = y_i = x_i^3 a_{i+1} + x_i^2 b_{i+1} + x_i c_{i+1} + d_{i+1}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def Spline(P):\n",
      "    \"\"\" Make a cubic spline that goes through the knots P \"\"\"\n",
      "    x = [1.*p[0] for p in P]\n",
      "    y = [1.*p[1] for p in P]\n",
      "   \n",
      "    n = len(P)-1\n",
      "    \n",
      "    A = zeros((n+1,)*2)\n",
      "    B = 3. * ones(n+1)\n",
      "    for i in xrange(1,n):\n",
      "        A[i, i-1:i+2] = [1./(x[i] - x[i-1]), 2.*(1./(x[i]-x[i-1]) + 1./(x[i+1]-x[i])), 1./(x[i+1]-x[i])]\n",
      "    \n",
      "    A[0,0:2] = [2./(x[1]-x[0]), 1./(x[1]-x[0])]\n",
      "    B[0] = 3. * ((y[1] - y[0]) / (x[1] - x[0])**2.)\n",
      "    A[n,n-1:] = [1./(x[n] - x[n-1]), 2./(x[n]-x[n-1])]\n",
      "    B[n] = 3. * ((y[n] - y[n-1]) / (x[n] - x[n-1])**2.)\n",
      "    \n",
      "    k = solve(A, B)[-1::-1]\n",
      "\n",
      "    \n",
      "    def p(xx):\n",
      "        for i in xrange(n):\n",
      "            if xx >= x[i] and xx <= x[i+1]:\n",
      "                break\n",
      "        if (i >= n):\n",
      "            i = n-1\n",
      "            \n",
      "        t = (xx - x[i]) / (x[i+1] - x[i])\n",
      "        a = k[i]*(x[i+1] - x[i]) - (y[i+1] - y[i])\n",
      "        b = -k[i+1]*(x[i+1]-x[i]) - (y[i+1] - y[i])\n",
      "        return (1. - t)*y[i] + t*y[i+1] + t*(1.-t)*(a*(1.-t)+b*t)\n",
      "        \n",
      "    return p\n",
      "                "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 216
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "points = [(-1,0),(0,-1),(1,0)]\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 245
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p1 = Spline(points)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 246
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = linspace(-1,1)\n",
      "plot(x, map(p1, x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 247,
       "text": [
        "[<matplotlib.lines.Line2D at 0x66545d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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1zU3OUaOC+0bv22+baZ6XX7adRIKRxxvh8/PzcblcALhcLvLz84tdU7duXR56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\nQsz/AujsWu8WjpNbAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x63b66d0>"
       ]
      }
     ],
     "prompt_number": 247
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x[:-1], diff(map(p1, x)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 248,
       "text": [
        "[<matplotlib.lines.Line2D at 0x6837810>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hoYHq6mqqq6txOp1s3rw5qAMBoFs3WLwY5s6F2lq7q5HDUU9BQpnHoTBnzhzW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\nERG3/wce7YPlUEmtNAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5bfe0d0>"
       ]
      }
     ],
     "prompt_number": 248
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x[:-2], diff(diff(map(p1, x))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 249,
       "text": [
        "[<matplotlib.lines.Line2D at 0x690d350>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x5886750>"
       ]
      }
     ],
     "prompt_number": 249
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 249
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}