finitediff.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finite Difference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy import init_printing, pprint, latex \n",
    "from sympy import binomial, Function, factorial, Symbol, symbols, Sum, S  \n",
    "# S is for SingletonRegistry\n",
    "from sympy import Rational as Rat\n",
    "from sympy import Matrix, solve_linear_system"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "EY : 20160627 I have no idea what `sympy.SingletonRegistry` means despite the [documentation for SingletonRegistry](http://docs.sympy.org/0.7.1/_modules/sympy/core/singleton.html), which is sparse at best.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "init_printing() # LaTeX printing "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = Symbol('n',integer=True,positive=True)\n",
    "i = Symbol('i',integer=True,positive=True)\n",
    "j = Symbol('j',integer=True,positive=True)\n",
    "x,h,x_0 = symbols('x h x_0',real=True)\n",
    "f = Function('f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "120"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "C_n = Sum((-1)**i*binomial(n,i),(i,1,n))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.00000000000000"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(C_n.subs(n,3)).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "cf. From *stackexchange* [How to do a symbolic taylor expansion of an unknown function $f(x)$ using sympy](http://stackoverflow.com/questions/16869587/how-to-do-a-symbolic-taylor-expansion-of-an-unknown-function-fx-using-sympy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$f{\\left (x_{0} \\right )} + \\left(x - x_{0}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\frac{1}{2} \\left(x - x_{0}\\right)^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(\\left(x - x_{0}\\right)^{3}; x\\rightarrowx_{0}\\right)$$"
      ],
      "text/plain": [
       "                                                ⎛  2        ⎞│                \n",
       "                                              2 ⎜ d         ⎟│                \n",
       "                                      (x - x₀) ⋅⎜────(f(ξ₁))⎟│                \n",
       "                                                ⎜   2       ⎟│                \n",
       "                 ⎛ d        ⎞│                  ⎝dξ₁        ⎠│ξ₁=x₀    ⎛      \n",
       "f(x₀) + (x - x₀)⋅⎜───(f(ξ₁))⎟│      + ───────────────────────────── + O⎝(x - x\n",
       "                 ⎝dξ₁       ⎠│ξ₁=x₀                 2                         \n",
       "\n",
       "            \n",
       "            \n",
       "            \n",
       "            \n",
       "  3        ⎞\n",
       "₀) ; x → x₀⎠\n",
       "            "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).series(x, x0=x_0, n=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                                              ⎛  2        ⎞│                  \n",
      "                                            2 ⎜ d         ⎟│                  \n",
      "                                    (-h + x) ⋅⎜────(f(ξ₁))⎟│                  \n",
      "                                              ⎜   2       ⎟│                  \n",
      "                ⎛ d        ⎞│                 ⎝dξ₁        ⎠│ξ₁=h    ⎛        3\n",
      "f(h) + (-h + x)⋅⎜───(f(ξ₁))⎟│     + ──────────────────────────── + O⎝(-h + x) \n",
      "                ⎝dξ₁       ⎠│ξ₁=h                2                            \n",
      "\n",
      "        \n",
      "        \n",
      "        \n",
      "        \n",
      "       ⎞\n",
      "; x → h⎠\n",
      "        \n"
     ]
    }
   ],
   "source": [
    "pprint( f(x).series(x, x0=h, n=3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'f{\\\\left (h \\\\right )} + \\\\left(- h + x\\\\right) \\\\left. \\\\frac{d}{d \\\\xi_{1}} f{\\\\left (\\\\xi_{1} \\\\right )} \\\\right|_{\\\\substack{ \\\\xi_{1}=h }} + \\\\frac{1}{2} \\\\left(- h + x\\\\right)^{2} \\\\left. \\\\frac{d^{2}}{d \\\\xi_{1}^{2}}  f{\\\\left (\\\\xi_{1} \\\\right )} \\\\right|_{\\\\substack{ \\\\xi_{1}=h }} + \\\\mathcal{O}\\\\left(\\\\left(- h + x\\\\right)^{3}; x\\\\rightarrowh\\\\right)'"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(f(x).series(x, x0=h, n=3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "FW = f(x+h).series(x+h, x0=x_0, n=3)\n",
    "FW = FW.subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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KZIg35l1f0kvjr1kgDPStZwhVZrLLj6rK2A6ODBQXHwHZuklULjbgtBbU0Da7d2pzAHnB\nxif+Yd4HWF4qP2gBcD2ywhO2MvBfDETI/tKwymgr2t07R8wqqkyK+Od4H2F76fyQBWBrT78+xcde\nxBFWGZjrusOXcKEz1axjpqBO/hVbklLE/yrvk8W+1P+IBR7vXX2YKqwylmHD4b/N9ORZCfG/yPtk\nsS/1P2CBSXz7bhv3IR430w8PbIz+AGGkkCf+N3kj/+v3Sy3wkHuvv5D81xL/Qlt/mvJ/AafWciCV\npoQ1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$f{\\left (x_{0} \\right )} + h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\frac{h^{2}}{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "                                  ⎛  2        ⎞│                              \n",
       "                                2 ⎜ d         ⎟│                              \n",
       "                               h ⋅⎜────(f(ξ₁))⎟│                              \n",
       "                                  ⎜   2       ⎟│                              \n",
       "          ⎛ d        ⎞│           ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 3    2        2    \n",
       "f(x₀) + h⋅⎜───(f(ξ₁))⎟│      + ────────────────────── + O⎝h  + h ⋅x + h⋅x  + x\n",
       "          ⎝dξ₁       ⎠│ξ₁=x₀             2                                    \n",
       "\n",
       "                   \n",
       "                   \n",
       "                   \n",
       "                   \n",
       "3                 ⎞\n",
       " ; (h, x) → (0, 0)⎠\n",
       "                   "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FW"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAA3CAMAAADZn0ObAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }}$$"
      ],
      "text/plain": [
       "  ⎛ d        ⎞│     \n",
       "h⋅⎜───(f(ξ₁))⎟│     \n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FW.args[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAqIAAAA4CAMAAAARvWrNAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nXX47Hv+N8v8L871brvSNR/kAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 f{\\left (x_{0} \\right )} + h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "             ⎛  2        ⎞│                                                 \n",
       "           2 ⎜ d         ⎟│         ⎛ 3    2        2    3                 ⎞\n",
       "2⋅f(x₀) + h ⋅⎜────(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "             ⎜   2       ⎟│                                                 \n",
       "             ⎝dξ₁        ⎠│ξ₁=x₀                                            "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(f(x+h).series(x+h,x0=x_0,n=3)+f(x-h).series(x-h,x0=x_0,n=3)).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAq0AAAA4CAMAAADgtjFAAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nvqK1KPlZBP4H5FxbhIh7T7gAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$4 f{\\left (x_{0} \\right )} + 5 h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "               ⎛  2        ⎞│                                                 \n",
       "             2 ⎜ d         ⎟│         ⎛ 3    2        2    3                 ⎞\n",
       "4⋅f(x₀) + 5⋅h ⋅⎜────(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "               ⎜   2       ⎟│                                                 \n",
       "               ⎝dξ₁        ⎠│ξ₁=x₀                                            "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "( f(x+h).series(x+h,x0=x_0,n=3) + f(x-h).series(x-h,x0=x_0,n=3) + \\\n",
    "f(x+2*h).series(x+2*h,x0=x_0,n=3) + f(x-2*h).series(x-2*h,x0=x_0,n=3) ).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAk0AAAA3CAMAAAAmAo7kAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$6 h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "    ⎛ d        ⎞│         ⎛ 3    2        2    3                 ⎞\n",
       "6⋅h⋅⎜───(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "    ⎝dξ₁       ⎠│ξ₁=x₀                                            "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "( f(x+h).series(x+h,x0=x_0,n=3) - f(x-h).series(x-h,x0=x_0,n=3) + \\\n",
    "f(x+2*h).series(x+2*h,x0=x_0,n=3) - f(x-2*h).series(x-2*h,x0=x_0,n=3) ).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "finite-difference for the 1st order derivative ($n=1$), to the $p$th-order, $p=8$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA4sAAAA3CAMAAACSLS/CAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{8} + h^{7} x + h^{6} x^{2} + h^{5} x^{3} + h^{4} x^{4} + h^{3} x^{5} + h^{2} x^{6} + h x^{7} + x^{8}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "  ⎛ d        ⎞│         ⎛ 8    7      6  2    5  3    4  4    3  5    2  6    \n",
       "h⋅⎜───(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  + h\n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀                                                          \n",
       "\n",
       "  7    8                 ⎞\n",
       "⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "                          "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 8\n",
    "( Rat(4,5)*(f(x+h).series(x+h,x0=x_0,n=p) - f(x-h).series(x-h,x0=x_0,n=p)) + \\\n",
    "Rat(-1,5)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)-f(x-2*h).series(x-2*h,x0=x_0,n=p)) + \\\n",
    "Rat(4,105)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)-f(x-3*h).series(x-3*h,x0=x_0,n=p)) + \\\n",
    "Rat(-1,280)*(f(x+4*h).series(x+4*h,x0=x_0,n=p)-f(x-4*h).series(x-4*h,x0=x_0,n=p)) ).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAkklEQVQYGWNgYGAQAmIgMAGTrClgqmIKmFoA\npjgKwNRWBjC1AEzxCICpXQxg6uzdu9+ugnVAtDN8AXOW/L8Bpski/oPAB6K0Tt0gdACoMP//L5Dy\nWVcLQNQFEAGnbrQqgnjeDPUPQDQDiwGYYvrOwMA7gYHrHwMD2wQGpt8MDEwMDMwTGBjYFRjaQMYU\nrdVmYAAALnIpDsFeUO4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{4}{5}$$"
      ],
      "text/plain": [
       "4/5"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rat(4,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### *centered difference approximation*   \n",
    "  \n",
    "$$-i_{\\text{min}} = (d+p-1)/2 $$  \n",
    "$$i_{\\text{max}} = \\lceil (d+p-1)/2 \\rceil $$\n",
    "\n",
    "cf. [Derivative Approximation by Finite Differences\n",
    "David Eberly](http://math.nyu.edu/~atm262/fall06/compmethods/a1/DerivativesApproximationByFiniteDifferences.pdf)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAyBAMAAAAAZPCUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiUSZq1TvELvdZiIy\nds1Wk1T5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACbUlEQVQ4EX1UTWsTQRh+l2wyySbZtPQHdH9B\nG1DEW/dQ8CDSehFEYYMW/IISEHrNiidBMKVS8da7YAOCFw+t+AMSELyINOBBEJTYDyrWou/H7jS7\nmd0X9p3nfZ5nZ2dmZwYAw/IwZYXytTKjkQnMxaTlxyjVPpy+jIwzjOhr7ZQeleo+vO0hfiS1umW2\ngd0Cu4na2jIbyoRN0RiCNUKh2GI1aJtMyHVCqP8lbZ0NG5wNqYT97RMf9DA5JwYLU/hR9w+hCo2s\nEhI0xg+4ekSCc4yp7xE0hvPlPY9PnaJ8wWiJSJt6AviJzwqjjOT6LOz1oMYjNdqcB9CnmQLsbEE9\nc7pQ/6a+SwclH8otgaY8c84TutyFUiiwRmuUFbiSnUjPWUZcwH3oD6STT2FWZ8jXTiFYZr36Os+n\nDmB7l32Wm+eDQ8AlpHiV7/sNs+xTg3zfKiy0qTsL8n33YIlscNPk+0cxYn1TfGrK5GOHpE35bnFx\ncWllMMan4ctoHqD/X9oh9Srs7QpqhNAZCjTku7DtMW09O9yqvDA4hDqAQHxc3kj53lx8Koz6pfcB\nEVNJH95osx5TuA8aTS2q1IztNpRarBaPoDLUPkcjAaUu2HwuwTqBwiilnpXuceyzR3LWz7QUqsj5\nxcNR45shJetyXqa2gyPPPefP5Q3apAv65UlQaAr3BJt5b1KPmY8C+B7iyy0Wkm2xCe+IqdNs+HJL\n6nH1AeAzYbw2MG5TMkV1ffp8l4QAp4G5TdkQLm579n1l0W0aPONU8Q5XVfmH40oSr3lSX0/SE9Xj\niCn4E9I44YRxdSUGxnZOswVPw0mgLhH3Hy5bhGA7Q1NEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( 4, \\quad \\frac{9}{2}\\right )$$"
      ],
      "text/plain": [
       "(4, 9/2)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 2 \n",
    "p = 8\n",
    "sympy.floor(Rat(d+p-1,2)), Rat((d+p-1),2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First order derivatives $d=1$; let's aim for $-i_{\\text{min}} = i_{\\text{max}} = 2$ so we only need 2 neighbor values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAyBAMAAAAAZPCUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n2GFOGiS8pqpmFdC2kylr+ynMA0e8ecB/lNmE/d/xt1IAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( 2, \\quad \\frac{5}{2}\\right )$$"
      ],
      "text/plain": [
       "(2, 5/2)"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 1 \n",
    "p = 5\n",
    "sympy.floor(Rat(d+p-1,2)), Rat((d+p-1),2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAyBAMAAAAAZPCUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n2GFOGiS8pqpmFdC2kylr+ynMA0e8ecB/lNmE/d/xt1IAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( 2, \\quad \\frac{5}{2}\\right )$$"
      ],
      "text/plain": [
       "(2, 5/2)"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 2 \n",
    "p = 4\n",
    "sympy.floor(Rat(d+p-1,2)), Rat((d+p-1),2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "C_1,C_2,C_3,C_4 = symbols('C_1 C_2 C_3 C_4', integer=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From `sympy` documentation, [Matrices (linear algebra), Basic Manipulation](http://docs.sympy.org/0.7.2/modules/matrices/matrices.html), I learned to construct matrices with the `lambda` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAABLCAMAAADQ8jKCAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 8 & 27 & 64\\\\1 & 32 & 243 & 1024\\\\1 & 128 & 2187 & 16384\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   8    27    64  ⎤\n",
       "⎢                   ⎥\n",
       "⎢1  32   243   1024 ⎥\n",
       "⎢                   ⎥\n",
       "⎣1  128  2187  16384⎦"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(3,4, lambda i,j: (j+1)**(2*(i+1)+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAZBAMAAAAfwGfXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 4 & 6 & 8\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[2  4  6  8]"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix(1,4,lambda i,j: 2*(j+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "C_nu_d1p6 = sympy.zeros(4,5) # d = 1, p = 6\n",
    "C_nu_d1p6[:3,:4] = Matrix(3,4,lambda i,j: (j+1)**(2*(i+1)+1))\n",
    "C_nu_d1p6[3,:4] = Matrix(1,4,lambda i,j: 2*(j+1))\n",
    "C_nu_d1p6[3,4] = Rat(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAABkCAMAAACGl+2PAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 8 & 27 & 64 & 0\\\\1 & 32 & 243 & 1024 & 0\\\\1 & 128 & 2187 & 16384 & 0\\\\2 & 4 & 6 & 8 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   8    27    64    0⎤\n",
       "⎢                      ⎥\n",
       "⎢1  32   243   1024   0⎥\n",
       "⎢                      ⎥\n",
       "⎢1  128  2187  16384  0⎥\n",
       "⎢                      ⎥\n",
       "⎣2   4    6      8    1⎦"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nu_d1p6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaEAAAAyBAMAAAAHA3moAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dIma7q80y\ndlRAyO8oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGE0lEQVRoBd1ZXWgcVRQ+s5vsJPszHSutmFKy\nWkSNhGwbKzSK2ReffEgDtYoVEkFSsEhWCwHxIUul1B9K2pdC82JKVEJEs/jgg1g6UPClD9lSiihI\nVkpRoZhoazUbdT3n3pm9M8ns7L1XE9ocyNxz7znn++63M3PvzAQAbWuOjne9vZn3JLQcLHjuXd22\nLdvu/CeyjYWkN+D0/W8cYy9zIeZyY0GQOBURDA9d936p8PDaUQ0OAWINCt+scj/5qxhb411VVWRM\nz6kqUubwzbKjzz/7foeFtpR8Gatc831VRQALiop0OMQ0M35FA0UW6JwS8dVeLLX+inQ4xDwDisZz\nLBCl6MQGKNLhaKDIlRKhyCivvyItDm1FMVh/RVoc2opubIAiLQ5dRUZ2/RXpcegqsmZn514si2o5\nT2311uMQMwmsdc1XBoDW9V+9dTj+g6ItG6BIg0NfUeyDalFUS3l9Q1+qlWhwiInEP1qZEb3gVWd8\nOztjZkU0yts5OVt8IyphbUwFXlQrEgUVXS2CtbcowCK8zAsA1/BPxRTgBawqUUDR/fRu0V8QaI09\n44wNEP+jcUJIRAFeVCsTuYrms4RxAWcJz5DX1FL0uNuyv2meP0EBXpQpE7lrzEQeMeL/ENBFOjS1\n/gqmqL14qsALfmWiFP+ZRwhifJiO99GBLDHFmtCDeZuGjYIXfNVzIloFeIGiTmT9SdUxdkP0lDlS\nusTabbzhY6uOsb/5QGzfbnKMQ6viYV0PvrvPpnAUvCj3iABO0aAM0VwBEx/KU/qoQ0dI9w2zNuqQ\n+ItHH4ZPotICMRc+XoyzyQViDTseEVh040pZ8gDAjnMsdYj9djZYw77KLTUy3wBzWxm+AYdhnn4R\nbjtGyL4CYCVL3rDXuvDxbJsvxOFZQWiVRwRbfYqML4joiAPh1dfLsJ1dOjBK3JYTVOTNJ9gm2KRu\nwCswng9GGvc8eFA6Ry4RXPYpaszBIl+XAfY65LKdKI6ihqkXaXwnymHOoh2Z6At68LDN8Y02cT2i\n9A/SiuJ41UH8eQIez+LhASlF5m+YGiMxr+GfnHnwmfNy+SzLI4oZ0op6HKo8Qgda967hLN1zFFiM\njBXKELZYBIuu1RidpwZLUGdJ5KPnwbtXXQBeJDYgOsEVSax1LXwZXsgT5r2T09S6ioL70QidEGHG\nvtnHqfc2HwrdjxL0UOUzFx5MtpMH4UVaKJFRds9RKJEoRq99iXX5U5AbCd5HqcvppzFgFdxooGnJ\ntjiBAd45b1+soPdjSAjvIpPtgf6YseeYA1FEVm/vzaK/wvXN3rMOGN/vygODoFH3Y2pnzpdu7fd1\noLVWy2I/5h+r+5d++dmud4QzVKP7DLJiRHiJQmZQ9PCnwl6qYhyOJgIz9D7qgNgtwMXsMw5BuIFn\nb05kPlEt+yhTXd9Qb6dvSLhDa7cqCn7+aQGPhh+mXpPu7qKgZ+yj9XGAxyCSCN6tVrwSX3sO4Dn4\nCeAtYBAUCVHkKyA3yftF3sgdp1ham1QyffaYpF1Ah+ggwJzzKMB2DkGE0oqkZuclTXmOREuKqgAD\nFVeRRIlIwReUOWfgNDzCISggoeidK46AkPNmTj4ll4hZqMi4iYrKSQ0irB+1k7VjRQ5BpM0Vtdvm\nLcpUsdMKD0eoiLbR+awOEb51Ym3Pis0haI7NFWHSGRU1PLd9WLaGFOE5YvuHBhFua9aHowfqEHKK\n5D49BCTIf4KoX3VYr0HUC/AeZH6/h124NIXm5whfgHrylCpvyRJtE5KGiui2XqxoEOFdmAPAh9L5\nCoMgyuaK8PW035acnZvWWoJ46IYYBkOKzgJM2BpE+BwK5utLuMUUGQThN1eEPwJ7jg2bTYMx3MQT\npQaxNcOkCLfHJ0GDiL7dZAp4jtoLDILAmys6Cm1YomRmDo4XZCtIUaJivAQaRDD24K6P4VkbTnII\n4nQf6FxhYbMw90zbYeNRY0e7dkeF/TH20dro/s4BHaILtdptSE/TkypBkLn/WcYrf7OY+xrRjqd+\nkxh/hYU0Luebw4xlV8dCeXMIgrFBV0hbVfn2vyN/AutQXUfH1B05Q9VJXSpSxb9/9M4nbGZ5cgAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{4}{5}, \\quad C_{2} : - \\frac{1}{5}, \\quad C_{3} : \\frac{4}{105}, \\quad C_{4} : - \\frac{1}{280}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 4/5, C₂: -1/5, C₃: 4/105, C₄: -1/280}"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve_linear_system( C_nu_d1p6, C_1,C_2,C_3,C_4 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also, another way is this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAABkCAMAAAAmCIVhAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n//XYttU/JRRoWcG5BxkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 8 & 27 & 64\\\\1 & 32 & 243 & 1024\\\\1 & 128 & 2187 & 16384\\\\2 & 4 & 6 & 8\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   8    27    64  ⎤\n",
       "⎢                   ⎥\n",
       "⎢1  32   243   1024 ⎥\n",
       "⎢                   ⎥\n",
       "⎢1  128  2187  16384⎥\n",
       "⎢                   ⎥\n",
       "⎣2   4    6      8  ⎦"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nuM_d1p6 = sympy.zeros(4,4) # d = 1, p = 6\n",
    "C_nuM_d1p6[:3,:4] = Matrix(3,4,lambda i,j: (j+1)**(2*(i+1)+1))\n",
    "C_nuM_d1p6[3,:4] = Matrix(1,4,lambda i,j: 2*(j+1))\n",
    "C_nuM_d1p6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAABmCAMAAADcfdxNAAAAflBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{61}{90} & \\frac{29}{360} & - \\frac{1}{360} & \\frac{4}{5}\\\\\\frac{169}{360} & - \\frac{13}{180} & \\frac{1}{360} & - \\frac{1}{5}\\\\- \\frac{1}{10} & \\frac{1}{40} & - \\frac{1}{840} & \\frac{4}{105}\\\\\\frac{7}{720} & - \\frac{1}{360} & \\frac{1}{5040} & - \\frac{1}{280}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-61      29                  ⎤\n",
       "⎢────    ───    -1/360   4/5  ⎥\n",
       "⎢ 90     360                  ⎥\n",
       "⎢                             ⎥\n",
       "⎢ 169    -13                  ⎥\n",
       "⎢ ───    ────   1/360    -1/5 ⎥\n",
       "⎢ 360    180                  ⎥\n",
       "⎢                             ⎥\n",
       "⎢-1/10   1/40   -1/840  4/105 ⎥\n",
       "⎢                             ⎥\n",
       "⎣7/720  -1/360  1/5040  -1/280⎦"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nuM_d1p6.inv() # and then read off the last column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAA0BAMAAACOQoCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrRcvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{1}{3} & \\frac{2}{3}\\\\\\frac{1}{6} & - \\frac{1}{12}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1/3   2/3 ⎤\n",
       "⎢           ⎥\n",
       "⎣1/6   -1/12⎦"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix([[1,8],[2,4]]).inv() # d = 1, p=2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "C_nu_d1p4 = sympy.zeros(3,4) # d = 1, p = 4\n",
    "C_nu_d1p4[:2,:3] = Matrix(2,3,lambda i,j: (j+1)**(2*(i+1)+1))\n",
    "C_nu_d1p4[2,:3] = Matrix(1,3,lambda i,j: 2*(j+1))\n",
    "C_nu_d1p4[2,3] = Rat(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAABLCAMAAABKveUfAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nN30rONiU/+mXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 8 & 27 & 0\\\\1 & 32 & 243 & 0\\\\2 & 4 & 6 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  8   27   0⎤\n",
       "⎢             ⎥\n",
       "⎢1  32  243  0⎥\n",
       "⎢             ⎥\n",
       "⎣2  4    6   1⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nu_d1p4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASUAAAAyBAMAAADo7dDTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVummPSSkrGMtJndVhoTTRZeI/AdTDVjCvqmyUQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{3}{4}, \\quad C_{2} : - \\frac{3}{20}, \\quad C_{3} : \\frac{1}{60}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 3/4, C₂: -3/20, C₃: 1/60}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve_linear_system( C_nu_d1p4, C_1,C_2,C_3 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwMAAAA3CAMAAABn/2Y3AAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nvTtMG1LdhBGTZyuFgkBBoCBQEPg5BP4FOa9fvnvUlsUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{6} + h^{5} x + h^{4} x^{2} + h^{3} x^{3} + h^{2} x^{4} + h x^{5} + x^{6}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "  ⎛ d        ⎞│         ⎛ 6    5      4  2    3  3    2  4      5    6        \n",
       "h⋅⎜───(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h⋅x  + x ; (h, x)\n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀                                                          \n",
       "\n",
       "         ⎞\n",
       " → (0, 0)⎠\n",
       "          "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 6\n",
    "( Rat(3,4)*(f(x+h).series(x+h,x0=x_0,n=p) - f(x-h).series(x-h,x0=x_0,n=p)) + \\\n",
    "Rat(-3,20)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)-f(x-2*h).series(x-2*h,x0=x_0,n=p)) + \\\n",
    "Rat(1,60)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)-f(x-3*h).series(x-3*h,x0=x_0,n=p))).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\boxed{   h \\frac{d}{dx}f(x) =  \\frac{3}{4} ( f(x+h) - f(x-h) ) + \\frac{-3}{20}( f(x+2h) - f(x-2h)) + \\frac{1}{60} (f(x+3h)-f(x-3h)) +O(h^6) = \\sum_{\\nu = 1}^3 C_{\\nu} (f(x+ \\nu h) - f(x - \\nu h) ) + O(h^6) }\n",
    "  $$\n",
    "\n",
    "with  \n",
    "\n",
    " * $C_1 = \\frac{3}{4}$  \n",
    " * $C_2 = \\frac{-3}{20}$  \n",
    " * $C_3 = \\frac{1}{60}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's generalize this procedure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_C_nu_dp(NU):\n",
    "    C_nu_dp = sympy.zeros(NU,NU+1)\n",
    "    C_nu_dp[:NU-1,:NU] = Matrix(NU-1,NU,lambda i,j: (j+1)**(2*(i+1)+1))\n",
    "    #C_nu_dp[NU-1,:NU]  = Matrix(1,NU,lambda i,j:2*(j+1))\n",
    "    C_nu_dp[NU-1,:NU]  = Matrix(1,NU,lambda i,j:(j+1))\n",
    "    C_nu_dp[NU-1,NU]   = Rat(1,2)\n",
    "\n",
    "    # sanity check\n",
    "    pprint( C_nu_dp)\n",
    "    Cnus = symbols(' '.join( ['C_'+str(nu)for nu in range(1,NU+1)]), real=True)\n",
    "    return solve_linear_system( C_nu_dp, *Cnus )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  8   0 ⎤\n",
      "⎢         ⎥\n",
      "⎣1  2  1/2⎦\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{C_1: 2/3, C_2: -1/12}"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_dp(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAf4AAAA3CAMAAADaIdjfAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nd07ib+v9b+9Pnz1d+Ns1dHf//wMVvD8G9lwL/gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{2} + h x + x^{2}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "  ⎛ d        ⎞│         ⎛ 2          2                 ⎞\n",
       "h⋅⎜───(f(ξ₁))⎟│      + O⎝h  + h⋅x + x ; (h, x) → (0, 0)⎠\n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀                                    "
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sanity check\n",
    "p=2\n",
    "(Rat(2,3)*(f(x+h).series(x+h,x0=x_0,n=p) - f(x-h).series(x-h,x0=x_0,n=p)) + \\\n",
    "Rat(-1,12)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)-f(x-2*h).series(x-2*h,x0=x_0,n=p))).simplify()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  8   27    0 ⎤\n",
      "⎢               ⎥\n",
      "⎢1  32  243   0 ⎥\n",
      "⎢               ⎥\n",
      "⎣1  2    3   1/2⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASUAAAAyBAMAAADo7dDTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVummPSSkrGMtJndVhoTTRZeI/AdTDVjCvqmyUQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{3}{4}, \\quad C_{2} : - \\frac{3}{20}, \\quad C_{3} : \\frac{1}{60}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 3/4, C₂: -3/20, C₃: 1/60}"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_dp(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1   8    27    64     0 ⎤\n",
      "⎢                        ⎥\n",
      "⎢1  32   243   1024    0 ⎥\n",
      "⎢                        ⎥\n",
      "⎢1  128  2187  16384   0 ⎥\n",
      "⎢                        ⎥\n",
      "⎣1   2    3      4    1/2⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaEAAAAyBAMAAAAHA3moAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{4}{5}, \\quad C_{2} : - \\frac{1}{5}, \\quad C_{3} : \\frac{4}{105}, \\quad C_{4} : - \\frac{1}{280}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 4/5, C₂: -1/5, C₃: 4/105, C₄: -1/280}"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_dp(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's attempt to further generalize this procedure for $d=1$.  \n",
    "\n",
    "Consider the following easy examples for $p=1$.  \n",
    "\n",
    "Now\n",
    "$$\n",
    "f(x \\pm h) = f(x) + (\\frac{ \\partial f}{ \\partial x}(x) )( \\pm h) + \\mathcal{O}(h^2)  \n",
    "$$  \n",
    "$$\n",
    "f(x+h) - f(x-h) = 2h \\frac{ \\partial f}{ \\partial x} + \\mathcal{O}(h^2) \\text{ or } \\frac{ \\partial f}{ \\partial x} = \\frac{ f(x+h) - f(x-h) }{2h} + \\mathcal{O}(h)\n",
    "$$\n",
    "\n",
    "Likewise, for so-called forward and backward difference operators:\n",
    "$$ \n",
    "\\frac{f(x+h) - f(x)}{h} = \\frac{ \\partial f}{ \\partial x}  \n",
    "$$\n",
    "$$ \n",
    "\\frac{f(x) - f(x-h)}{h} = \\frac{ \\partial f}{ \\partial x}  \n",
    "$$\n",
    "Conjecture that \n",
    "$$\n",
    "\\boxed{\n",
    "\\frac{ \\partial f}{ \\partial x} = \\sum^N_{\\nu = 1} C_{\\nu} (f(x+ \\nu h) - f(x-\\nu h) ) \\left( \\frac{1}{h} \\right)\n",
    "}\n",
    "$$\n",
    "\n",
    "and so doing the Taylor expansion,\n",
    "$$\n",
    "f(x \\pm \\nu h) =\\sum_{j=0}^{p} \\frac{ f^{(j)}(x) }{j!} ( \\pm \\nu h)^j + \\mathcal{O}(h^{p+1}) = \\sum_{j'=0}^{p'} \\frac{ f^{(2j')}(x) }{ (2j')! } (\\nu h)^{2j'} + \\sum_{j'=0}^{p'} \\frac{ f^{(2j'+1) }(x) }{ (2j'+1)!} (\\nu h)^{2j'+1} (\\pm 1) + \\mathcal{O}(h^{p+1} )\n",
    "$$\n",
    "where $p := 2p'+1$.  \n",
    "\n",
    "Clearly, for $f(x+\\nu h) - f(x-\\nu h)$, all the \"even\" terms in the Taylor series expansion cancel each other out, leaving\n",
    "$$\n",
    "f(x+\\nu h) - f(x-\\nu h) = \\sum_{j'=0}^{p'} \\frac{ 2f^{(2j'+1)}(x) }{(2j'+1)!} (\\nu h)^{2j'+1} + O(h^{p+1}) = 2\\nu h \\frac{ \\partial f}{\\partial x} + \\sum_{j'=1}^{p'} \\frac{2(\\nu h)^{2j' +1} }{ (2j'+1)! } f^{(2j'+1) }(x) + \\mathcal{O}(h^{p+1} ) \n",
    "$$\n",
    "\n",
    "and so with our previous assumption (caveat),  \n",
    "\n",
    "$$\n",
    "\\sum_{\\nu =1}^{N} C_{\\nu} ( f(x+ \\nu h) - f(x-\\nu h)) \\frac{1}{h} = \\sum_{\\nu =1}^N C_{\\nu} \\left( 2\\nu \\frac{ \\partial f}{ \\partial x} + \\sum_{j'=1}^{p'} \\frac{ 2\\nu ^{2j'+1} }{ (2j'+1)! } f^{(2j+1) }(x) h^{2j'} \\right) + \\mathcal{O}(h^{p} )\n",
    "$$\n",
    "\n",
    "Then we obtain a system of linear equations that we'd want to solve:\n",
    "\n",
    "$$\n",
    "\\sum_{\\nu =1}^N 2C_{\\nu} \\nu = 1 \n",
    "$$  \n",
    "\n",
    "$$\n",
    "\\sum_{j'=1}^{p'} \\left( \\sum_{\\nu =1}^N C_{\\nu} \\frac{2\\nu^{2j'+1} }{ (2j'+1)! } \\right) f^{(2j+1) }(x) h^{2j'} = 0 \\text{ or }  \\left( \\sum_{\\nu =1}^N C_{\\nu} \\frac{ 2 \\nu^{2j'+1} }{ (2j'+1)! } \\right) = 0 \\qquad \\, \\forall \\, j' = 1 \\dots p'\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we confirm the `make_C_nu_dp(NU)` Python function for $d=1$.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Backward difference operator, in the case of $d=1$, first-order derivative\n",
    "\n",
    "For the backward difference operator, then consider the following:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{ \\partial f}{ \\partial x} = \\sum_{ \\nu =0}^N \\frac{1}{h} C_{\\nu} f(x-\\nu h) = \\sum_{\\nu = 0}^N \\frac{C_{\\nu} }{ h} \\left[ f(x) + \\left( \\frac{ \\partial f}{ \\partial x} \\right) (-\\nu h) + \\sum_{j=2}^p \\frac{ f^{(j)}(x) }{j!} (-\\nu h)^j \\right] + \\mathcal{O}(h^p)\n",
    "$$\n",
    "\n",
    "which leads to a system of linear equations:\n",
    "\n",
    "$$\n",
    "\\frac{1}{h} \\sum_{\\nu =0}^N C_{\\nu} = 0 \n",
    "$$\n",
    "\n",
    "$$ \n",
    "- \\sum_{\\nu =0}^N C_{\\nu } \\nu = 1\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\sum_{j=2}^p \\frac{(-h)^jf^{(j)}(x)}{ hj!} \\sum_{ \\nu =0}^N C_{\\nu} \\nu^j = 0\n",
    "$$\n",
    "\n",
    "and I didn't make it explicit before, and neither here, but you can divide both sides of the equation by constant values that *don't* depend on index $j$ (or $\\nu$) and obtain the system of linear equations to solve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1  1  1   0 ⎤\n",
      "⎢               ⎥\n",
      "⎢0  1  2  3   -1⎥\n",
      "⎢               ⎥\n",
      "⎢0  1  4  9   0 ⎥\n",
      "⎢               ⎥\n",
      "⎣0  1  8  27  0 ⎦\n",
      "(C₀, C₁, C₂, C₃)\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAAyBAMAAABCLhZvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dIma7q80y\ndlRAyO8oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFUUlEQVRoBdVZXWhcRRQ+d+/u3s3P7l6UVGwo\nWVNUFEK2xD4kL1mQPjeB+EfRxlqiWCRLAhWfsuCLbw0oPhTRqCglKuZJfKh4IeBLHrJERHzKQqlF\nENPUoqbVxjlnZu7ee3dm907SkN2BzJw55/vOfDt3du5JAsDaA0XsO6y9VZKCk8+WpdlBY2bHFWov\nFQ5P9tzgh3td/OLLnOns6DPkJlmMOj1mPxGnCFsV0wRCj3OHE7u3tQmOjrEYdVrI/gLZe5CfNkzh\n6xn3iJlf0SfI4mejTo/ZTyQ5BQNNllenlnpOVyg+sKSGofeg9bMlzM+P1L9QJOGHq986RyJMOqlf\nCD9U/c5jJ0ykE7ad9AM8WjL9AO2lPzXZwfqdMnTd62D9+e3O1p9ahPTfHbz/mQpsVQ9Ov/3Z3StA\nnekacfFPj7wXFypxvp7w/W/9snzFKUhQjPHY5eXKnBqnriqbEKJpYokJ6/+pArmTlWgi/Tz7PMA1\n9qNo6qqyCaEhRywxIf0PYzU9Xm7IpHNY77sAtvq7p6wqmxGii8QTI/SvF5D+PZMDp9CK13q2GS45\noQQrq8pmhGiWeGLyi8S7VGKDTW+R1Wgi/Xy8xmK9RR2gsapsQQgmiimmh+/eDFIXprF/CDts6SUa\n9J3zF8asskS8Lg0+NlaVLQghekwxuX+QlaAzPFzlGTY2aOxb4VNtn/hPhNZotM6EkIqqUhISoycQ\nGiGE2CDFDI25GNCK+Rq3j9d/sx4iIVPILZHRqkv/yxGrn6uRDVWlJDwOX6gZAa8QY1fsxYC30eye\nAuj/mPxn6ZO66RLg91K0/K7fpEuOKYJZAL9KDxutb2ZYu+Axs6GqlIRXYR13TTTKf1PO5CjE2IVM\nINSPuWeuAhfFodercIQeJ8yiI+cNePCSzNJ0TFPm38P6JUNVVUrCeVgoSZxuFGLYrbKog3D/D1WA\nkx7adPPbuDev8VCLnt/8RbV+VVXpE2DLbZFbimEn32sKtdn5Afs5xCwUWPcIrLvwBk5bNucWgySY\nkuD5kSxVVekTYiwgxED2O5lRPQ576L+AHd5F11zc/xdx2viVt+6iv962WLmBR4/rD18nVFXqCAn2\n0KL3T+RvKEKMPD+NYriMJL8DN0s4ffDyp2xk5/82zhT3/0z4sVujy08hUux/+P6nqlJDeBtZACFC\nGouXQONiABx6qyrEELbrJg28fuDs8P1DvsENj8ZcmUMiver8cIiakCwkvUCK+dEPcHYj4PLNPs+h\nN5Tv4IYz+ESNLPGHtwF6oDyWLUbv//6SvUKxBEdE++tRhz9XE9b++M31IWAtwXCJTQt1V91Kl7OT\n9ZlvzdNhZ9NQ/SnCx48HsqPvXehapNgxgQgP87NfhR31mZpwdne3DoFuF1IT7NtQDfh8s3foybI/\nqRubNaDiRam/DuNW0q+QK9FQq3kcQmoautmbPNMqVzB+qiSuBtX+B4Fod7HdOcDGimrUb9h44RZr\n//NXh+iaMVzBAJ6mSsSAAHCkQPA4+z/wLaRqRslNwacLpoy5V1yixNK/DbbqEjBdU48PvQj0sGAk\nMUWzOPrZf0cyt4Pc+23bxT1kfIYeQBz96RXIqF4ie1hUTflR7W7ifRPgoxrG4+jvnjjY/c8Vgekx\naruugf5edv4P8gpdBfjZSD3ACwCzZeTE2X/4BPpqhgsYwJ1zg2PTBniEvgM5XniKwkd8DE2WxMiX\nmsj9cPewXyCnDRMlR0ZLRBH/v0utGCZoF7gonLvMX4Dt8Qn4r1/Q+2d7yDFVYe0IxmbVlNoW+Iuy\nLsjccdtCkJmI3Blf9dElM2pboNcqKON/e01zkmAqI6cAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ C_{0} : \\frac{11}{6}, \\quad C_{1} : -3, \\quad C_{2} : \\frac{3}{2}, \\quad C_{3} : - \\frac{1}{3}\\right \\}$$"
      ],
      "text/plain": [
       "{C₀: 11/6, C₁: -3, C₂: 3/2, C₃: -1/3}"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NU = 3\n",
    "C_nueqns = sympy.zeros(NU+1,NU+2)\n",
    "C_nueqns[0,:NU+1] = Matrix(1,NU+1,lambda i,j: 1)\n",
    "C_nueqns[1,:NU+1] = Matrix(1,NU+1,lambda i,j:j)\n",
    "C_nueqns[1,NU+1] = -1\n",
    "if NU > 1:\n",
    "    C_nueqns[2:NU+1,:NU+1] = Matrix(NU-1,NU+1,lambda i,j: j**(i+2) )\n",
    "pprint(C_nueqns)\n",
    "Cnus = symbols(' '.join( ['C_'+str(nu)for nu in range(NU+1)]), real=True)\n",
    "pprint(Cnus)\n",
    "solve_linear_system( C_nueqns,*Cnus)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def make_backwardC_nu(NU):\n",
    "    C_nueqns = sympy.zeros(NU+1,NU+2)\n",
    "    C_nueqns[0,:NU+1]  = Matrix(1,NU+1,lambda i,j: 1)\n",
    "\n",
    "    C_nueqns[1,:NU+1]  = Matrix(1,NU+1,lambda i,j: j)\n",
    "    C_nueqns[1,NU+1]   = -1\n",
    "\n",
    "    if NU > 1:\n",
    "        C_nueqns[2:NU+1,:NU+1] = Matrix(NU-1,NU+1,lambda i,j: (j)**(i+2) )\n",
    "\n",
    "    # sanity check\n",
    "    pprint( C_nueqns)    \n",
    "        \n",
    "    Cnus = symbols(' '.join( ['C_'+str(nu)for nu in range(NU+1)]), real=True)\n",
    "    return solve_linear_system( C_nueqns, *Cnus )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1  0 ⎤\n",
      "⎢        ⎥\n",
      "⎣0  1  -1⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAAVBAMAAAAgMbgsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDXUa3Now9AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{0} : 1, \\quad C_{1} : -1\\right \\}$$"
      ],
      "text/plain": [
       "{C₀: 1, C₁: -1}"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_backwardC_nu(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1  1  0 ⎤\n",
      "⎢           ⎥\n",
      "⎢0  1  2  -1⎥\n",
      "⎢           ⎥\n",
      "⎣0  1  4  0 ⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAAAyBAMAAABbpfjJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left \\{ C_{0} : \\frac{3}{2}, \\quad C_{1} : -2, \\quad C_{2} : \\frac{1}{2}\\right \\}$$"
      ],
      "text/plain": [
       "{C₀: 3/2, C₁: -2, C₂: 1/2}"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_backwardC_nu(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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i2TR1N0F/9BWINYg87uo7gX+RDS8o5yww5TZYnxP7cq6/ivvlhvlHFfwfKa9v9C7FEkkA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} - \\frac{h^{3}}{3} \\left. \\frac{d^{3}}{d \\xi_{1}^{3}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{4} + h^{3} x + h^{2} x^{2} + h x^{3} + x^{4}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "                          ⎛  3        ⎞│                                      \n",
       "                        3 ⎜ d         ⎟│                                      \n",
       "                       h ⋅⎜────(f(ξ₁))⎟│                                      \n",
       "                          ⎜   3       ⎟│                                      \n",
       "  ⎛ d        ⎞│           ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 4    3      2  2      3    \n",
       "h⋅⎜───(f(ξ₁))⎟│      - ────────────────────── + O⎝h  + h ⋅x + h ⋅x  + h⋅x  + x\n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀             3                                            \n",
       "\n",
       "                   \n",
       "                   \n",
       "                   \n",
       "                   \n",
       "4                 ⎞\n",
       " ; (h, x) → (0, 0)⎠\n",
       "                   "
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sanity check\n",
    "f(x).series(x,x0=x_0,n=4).subs(x-x_0,0)\n",
    "f(x-h).series(x-h,x0=x_0,n=4).subs(x-x_0,0)\n",
    "f(x-2*h).series(x-2*h,x0=x_0,n=4).subs(x-x_0,0)\n",
    "\n",
    "p=4\n",
    "( Rat(3,2)*f(x).series(x,x0=x_0,n=p).subs(x-x_0,0) + \\\n",
    "  Rat(-2)*( f(x-h).series(x-h,x0=x_0,n=p) ).subs(x-x_0,0) + \\\n",
    "Rat(1,2)*( f(x-2*h).series(x-2*h,x0=x_0,n=p)).subs(x-x_0,0) )\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1  1  1   0 ⎤\n",
      "⎢               ⎥\n",
      "⎢0  1  2  3   -1⎥\n",
      "⎢               ⎥\n",
      "⎢0  1  4  9   0 ⎥\n",
      "⎢               ⎥\n",
      "⎣0  1  8  27  0 ⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAAyBAMAAABCLhZvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLsjccdtCkJmI3Blf9dElM2pboNcqKON/e01zkmAqI6cAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ C_{0} : \\frac{11}{6}, \\quad C_{1} : -3, \\quad C_{2} : \\frac{3}{2}, \\quad C_{3} : - \\frac{1}{3}\\right \\}$$"
      ],
      "text/plain": [
       "{C₀: 11/6, C₁: -3, C₂: 3/2, C₃: -1/3}"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_backwardC_nu(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} - \\frac{h^{4}}{4} \\left. \\frac{d^{4}}{d \\xi_{1}^{4}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{5} + h^{4} x + h^{3} x^{2} + h^{2} x^{3} + h x^{4} + x^{5}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "                          ⎛  4        ⎞│                                      \n",
       "                        4 ⎜ d         ⎟│                                      \n",
       "                       h ⋅⎜────(f(ξ₁))⎟│                                      \n",
       "                          ⎜   4       ⎟│                                      \n",
       "  ⎛ d        ⎞│           ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 5    4      3  2    2  3   \n",
       "h⋅⎜───(f(ξ₁))⎟│      - ────────────────────── + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + \n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀             4                                            \n",
       "\n",
       "                           \n",
       "                           \n",
       "                           \n",
       "                           \n",
       "   4    5                 ⎞\n",
       "h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "                           "
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sanity check\n",
    "p=5\n",
    "( Rat(11,6)*f(x).series(x,x0=x_0,n=p).subs(x-x_0,0) + \\\n",
    "  Rat(-3)*( f(x-h).series(x-h,x0=x_0,n=p) ).subs(x-x_0,0) + \\\n",
    "Rat(3,2)*( f(x-2*h).series(x-2*h,x0=x_0,n=p)).subs(x-x_0,0) + \\\n",
    "Rat(-1,3)*( f(x-3*h).series(x-3*h,x0=x_0,n=p)).subs(x-x_0,0) \n",
    ")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  1  1   1    1   0 ⎤\n",
      "⎢                     ⎥\n",
      "⎢0  1  2   3    4   -1⎥\n",
      "⎢                     ⎥\n",
      "⎢0  1  4   9   16   0 ⎥\n",
      "⎢                     ⎥\n",
      "⎢0  1  8   27  64   0 ⎥\n",
      "⎢                     ⎥\n",
      "⎣0  1  16  81  256  0 ⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAc8AAAAyBAMAAADILU1rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dIma7q80y\ndlRAyO8oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGLklEQVRoBd1aXWhcRRQ+d3ezd/Ozm6WiYiTk\n2qCiL4mNlaYPZkF8toW2Gqs2iqRqkCwVgj7tgg8KPiQPUrAPNlLUsqLJk+BTFgVfCmZNKSI+tBBE\nlGKaWqtJtes587O5Mztzc+9DbtYdyJ05Z75z5nz3Z2bOTgCw7Bmma7uWtwqSWepYUTbbsc5s5AWt\nWc/Gzx35oArd3/c8bgPshv6bqIPOvMgt3A2rZR8kbkBHve5ZEbvQ8WroMXOHGdTd5BZd61bLjwCO\nQvfDP1gBu9CRHQ87aN9BwWysykx6F62WxwAWql3W7l3pePOr0MNmBdGnysxkYN5quZRvPaJedKIl\nvqoEEMU7MJ3vevdi1XorYu/IFKMTFQwDiaauQ2fevRE7H+uA/bAzRNPzNORp67ixd3g7RHSEMRlr\nmR1FrrozRLP4GX8GMFSI/clZBtwDO0N0FdzicYCxvGXc2NW/VSp/fh52VLm8bD8Z9QxDtogPdSqs\n6zhw46EHCU90Zu/gJ/AGZJ4O7TsG4EboMcITXarXb4K771zLvLnI8aH6+ZBMkx/f4lD11XV+rJx3\nvZA+CBZoYEox+s9UyqfUAU7t/VBVMMkANKAUVWAwKtFLZcjtLyvWwUKggSHFyD4DsIp/vuIOw1rZ\nJ/OmAdiE0RWBwShE76GkLcp6GWhgSDGc0/gBJP9SIszeht4JRYOCCahjdDkwGBBElz0yw/07wJPU\nClkCDQwpRvc6Ok4dUrynjsDAoqJBwQTUMbocGAz0zjH8bAGr5G1qm74s0htKsIHXvCEdu4JecL3S\nSvOrawFqdooYHAx089s7STalCbreTRcqfIvL2+arZvCagjKkGO5NQjhFiRMGzktSIWsLUHYbay0Y\nPfrc32SVYJ/NUI17WFlh9Z2LXLRfpUEPQzq4ffIVQ4qR+Jf3J0YfoYYwcB9gks8UGsCL+PtGA+hH\nNLdlMNDJ+pqiX6D7e3+BOqerdIWMl5tnjW0vwqDn4IQBakgx0v9w3IO0d/YVPrxPIYEHYNSnDW6K\nYADW8kZg1xGAe9l9gxMMkU8XgOYMUXrrjSJVjVoYQG6ioQLny0ksU1VTitHB/DpwEpbp9jZKx+FG\nkzckcB/MbvVIz8ACurbVwVsyGGfIR5RHzwE/1+Au/u5MkyJXHajCC7xru6swUIhKG1OKkWbRXYWX\noVSQOMwYoJPNgg0NTg8CCFGeKNnj7c0O+Ij6fMK3NYD9VdKwFTRJd/sVP8DeFgZGoqYUg6+gw+jQ\n93r1rjcTbQCd5+yjaz0ymO8sRJP46kKS7dhLHjbvg+U8vK45sYjCwEgUTcZ1M/c6ahJ0w30DdMxB\nWt1CADSAP+FTCFlkMJ6F6FCVHE3RhWbg1Tw90edJbJq3QF/ZhYEgqs26ABvg3CJHWwUXzBx9Jgl6\nqmLWzZRhrWYD8le3yTPzqG0zRDCZIieqR5/iM/7lAtneceYc1viNsh/C9JUIAWnaI/oKN5BPVF1H\nWYoxSY9vqzijlUdJepuruMETI+8DWIBQukJQzTO3NgfTD5yoHn3nNWbFt4DCgTLrch07gMHmL1zU\nrv5ZV+3KFVWZSykvVdX0RiB+oaWyBiRR5DvGYFZHxg4oJmIjJn6iH2DvEgdkh9V1lE4v2AEMdnuK\nDynkDsmWXid0BZMv/P5rXuswA0/CJR2IdjLf8TQfQiypJuk5pleyF4EcHPRD2ekFLrRHsdepCYhS\nuY9tGvUI6leAUjhRr8umrM3Ar1c+lQBfLfIdczCQOfuFDwtwyU5UwQHQjxHsAAY3TVrXtmJ5W4QA\nhAYiXuQ74YJx34tEdIkOYERMrVDh9B2yJLojEUWv0/mQrmOANec71kHfiUoUD2BaphjyHVtsTi0q\nUX1dsnmOR9+U79iGTUBUovwAxuYvbn1TvmML4GpUotlhm6v49aZ8xxKF40UlSgcwFmexq035jiWI\nXKWy8GyNOk0bBs2I1lHcR2Vbhqgp39Fi9omIphKSKDuA8VnvapPlO6EjEL9zik2u4GuyZqcXS3QA\n0zKF5Tsho0mc3SwTVPyzRsdiSLv/LUzkZ534HbZ34T8wQM8f7U0TnA1B8DKbg9uX7Qwm1axkNvPt\nyxJ/DjveoNc3385EL5SJ3X+E2q2GqrtwMQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ C_{0} : \\frac{25}{12}, \\quad C_{1} : -4, \\quad C_{2} : 3, \\quad C_{3} : - \\frac{4}{3}, \\quad C_{4} : \\frac{1}{4}\\right \\}$$"
      ],
      "text/plain": [
       "⎧    25                                  ⎫\n",
       "⎨C₀: ──, C₁: -4, C₂: 3, C₃: -4/3, C₄: 1/4⎬\n",
       "⎩    12                                  ⎭"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_backwardC_nu(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5kSfxB5axIFAIFqLgUyEt/HqLe8bytDLJUAS2Hja7uN8M/Fp1K1e1AG+BIFiIJ0vkKuft\n+0kc6hqncMwLB6ZUnBKcwGgx2jAhMl9kYNMYijyXQ5HF3bwDuAOh3lQLXLcA9INesNB5WdIJ8EsS\nYTgjiFQbt+JzLv6PaJzXBxwatvMty9OCjuAuDfw0yMpQLRCzQBgsFKOM5wsnkAtNikQ6Lw6bZfyg\n+Cre1qyCMhcwEddDSixsjC4mpQeTHG4ybH4L8IPQKlm1wGcsIJyAbBkHPS/8XqGIP5wKOq+uGIku\nvlDVHG44Y/0u4BfQVpZqgTdYQETxtd190vs7gfPCOvEIP1wFQ91iw+Yitcjhbn4VeJHaVyHVAtYC\n82NXn68NnNeSNAUXrFDoi6kjuH8S+Iv1ruzVAsQCq/js+Lbs8c/Htf1s+mfC9+1kHnfzm8C/bbiq\n/x9jgTn34Z0frelfxf2j5qywClng/3sHgeizbaDqAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} - \\frac{h^{5}}{5} \\left. \\frac{d^{5}}{d \\xi_{1}^{5}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{6} + h^{5} x + h^{4} x^{2} + h^{3} x^{3} + h^{2} x^{4} + h x^{5} + x^{6}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "                          ⎛  5        ⎞│                                      \n",
       "                        5 ⎜ d         ⎟│                                      \n",
       "                       h ⋅⎜────(f(ξ₁))⎟│                                      \n",
       "                          ⎜   5       ⎟│                                      \n",
       "  ⎛ d        ⎞│           ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 6    5      4  2    3  3   \n",
       "h⋅⎜───(f(ξ₁))⎟│      - ────────────────────── + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + \n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀             5                                            \n",
       "\n",
       "                                   \n",
       "                                   \n",
       "                                   \n",
       "                                   \n",
       " 2  4      5    6                 ⎞\n",
       "h ⋅x  + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "                                   "
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sanity check\n",
    "p=6\n",
    "( Rat(25,12)*f(x).series(x,x0=x_0,n=p).subs(x-x_0,0) + \\\n",
    "  Rat(-4)*( f(x-h).series(x-h,x0=x_0,n=p) ).subs(x-x_0,0) + \\\n",
    "Rat(3)*( f(x-2*h).series(x-2*h,x0=x_0,n=p)).subs(x-x_0,0) + \\\n",
    "Rat(-4,3)*( f(x-3*h).series(x-3*h,x0=x_0,n=p)).subs(x-x_0,0) + \\\n",
    "Rat(1,4)*( f(x-4*h).series(x-4*h,x0=x_0,n=p)).subs(x-x_0,0) \n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### For $d=2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "C_nu_d2p4 = sympy.zeros(2,3) # d = 2, p = 4\n",
    "C_nu_d2p4[:1,:2] = Matrix(1,2,lambda i,j: (j+1)**(2*(i+2)))\n",
    "C_nu_d2p4[1,:2] = Matrix(1,2,lambda i,j: 2*(j+1)**2)\n",
    "C_nu_d2p4[1,2] = Rat(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAyBAMAAADICBpCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBw+GchwBp0QAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 16 & 0\\\\2 & 8 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  16  0⎤\n",
       "⎢        ⎥\n",
       "⎣2  8   1⎦"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nu_d2p4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAAyBAMAAAAegQJiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{2}{3}, \\quad C_{2} : - \\frac{1}{24}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 2/3, C₂: -1/24}"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve_linear_system( C_nu_d2p4, C_1,C_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "C_nu_d2p6 = sympy.zeros(3,4) # d = 2, p = 6\n",
    "C_nu_d2p6[:2,:3] = Matrix(2,3,lambda i,j: (j+1)**(2*(i+2)))\n",
    "C_nu_d2p6[2,:3] = Matrix(1,3,lambda i,j: 2*(j+1)**2)\n",
    "C_nu_d2p6[2,3] = Rat(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAABLCAMAAABKveUfAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nqx0G9QdSHTt1GvgPQAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 16 & 81 & 0\\\\1 & 64 & 729 & 0\\\\2 & 8 & 18 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  16  81   0⎤\n",
       "⎢             ⎥\n",
       "⎢1  64  729  0⎥\n",
       "⎢             ⎥\n",
       "⎣2  8   18   1⎦"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_nu_d2p6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATEAAAAyBAMAAADGqPFBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{3}{4}, \\quad C_{2} : - \\frac{3}{40}, \\quad C_{3} : \\frac{1}{180}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 3/4, C₂: -3/40, C₃: 1/180}"
      ]
     },
     "execution_count": 170,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve_linear_system( C_nu_d2p6, C_1,C_2,C_3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAApMAAAA4CAMAAACWjIPLAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nuwUW+u7FbzdGa7+3wL+I0lezZbr9XQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{h^{2}}{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{4} + h^{3} x + h^{2} x^{2} + h x^{3} + x^{4}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                         \n",
       " 2 ⎜ d         ⎟│                                                         \n",
       "h ⋅⎜────(f(ξ₁))⎟│                                                         \n",
       "   ⎜   2       ⎟│                                                         \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 4    3      2  2      3    4                 ⎞\n",
       "────────────────────── + O⎝h  + h ⋅x + h ⋅x  + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "          2                                                               "
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 4\n",
    "( Rat(2,3)*(f(x+h).series(x+h,x0=x_0,n=p) + f(x-h).series(x-h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(-1,24)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)+f(x-2*h).series(x-2*h,x0=x_0,n=p) -2*f(x_0))).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAxIAAAA4CAMAAABexT40AAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{h^{2}}{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{6} + h^{5} x + h^{4} x^{2} + h^{3} x^{3} + h^{2} x^{4} + h x^{5} + x^{6}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                             \n",
       " 2 ⎜ d         ⎟│                                                             \n",
       "h ⋅⎜────(f(ξ₁))⎟│                                                             \n",
       "   ⎜   2       ⎟│                                                             \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 6    5      4  2    3  3    2  4      5    6      \n",
       "────────────────────── + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h⋅x  + x ; (h, \n",
       "          2                                                                   \n",
       "\n",
       "            \n",
       "            \n",
       "            \n",
       "            \n",
       "           ⎞\n",
       "x) → (0, 0)⎠\n",
       "            "
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 6\n",
    "( Rat(3,4)*(f(x+h).series(x+h,x0=x_0,n=p) + f(x-h).series(x-h,x0=x_0,n=p)-2*f(x_0)) + \\\n",
    "Rat(-3,40)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)+f(x-2*h).series(x-2*h,x0=x_0,n=p)-2*f(x_0)) + \\\n",
    "Rat(1,180)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)+f(x-3*h).series(x-3*h,x0=x_0,n=p)-2*f(x_0))).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Central difference operator, in the case of $d=2$, 2nd-order derivative\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's attempt to further generalize this procedure for $d=2$.  \n",
    "\n",
    "Consider the following easy examples for $p=1$.  \n",
    "\n",
    "Now\n",
    "$$\n",
    "f(x \\pm h) = f(x) + (\\frac{ \\partial f}{ \\partial x}(x) )( \\pm h) + ( \\frac{1}{2} \\frac{ \\partial^2 f}{ \\partial x^2}(x) )h^2 + \\mathcal{O}(h^2)  \n",
    "$$  \n",
    "$$\n",
    "f(x+h) + f(x-h) = 2f(x) + h^2 \\frac{ \\partial^2 f}{ \\partial x^2} + \\mathcal{O}(h^3) \\text{ or } \\frac{ \\partial^2 f}{ \\partial x^2} = \\frac{ f(x+h) + f(x-h) - 2f(x) }{h^2} + \\mathcal{O}(h)\n",
    "$$\n",
    "\n",
    "And so\n",
    "\n",
    "$$\n",
    "f(x \\pm \\nu h) = f(x) + (\\frac{ \\partial f}{ \\partial x}(x) )( \\pm \\nu h) + (\\frac{1}{2} \\frac{ \\partial^2 f}{ \\partial x^2}(x) ) \\nu^2 h^2 + \\mathcal{O}(h^2)  \n",
    "$$  \n",
    "$$\n",
    "f(x+\\nu h) + f(x-\\nu h) = 2f(x) + \\nu^2 h^2 \\frac{ \\partial^2 f}{ \\partial x^2} + \\mathcal{O}(h^3) \\text{ or } \\frac{ \\partial^2 f}{ \\partial x^2} = \\frac{ f(x+\\nu h) + f(x-\\nu h) - 2f(x) }{\\nu^2 h^2} + \\mathcal{O}(h)\n",
    "$$\n",
    "\n",
    "\n",
    "Conjecture that \n",
    "$$\n",
    "\\boxed{\n",
    "\\frac{ \\partial^2 f}{ \\partial x^2} = \\sum^N_{\\nu = 1} C_{\\nu} (f(x+ \\nu h) + f(x-\\nu h) -2f(x) ) \\left( \\frac{1}{h^2} \\right)\n",
    "}\n",
    "$$\n",
    "(yes, the $\\frac{1}{\\nu^2}$ factor was swallowed into $C_{\\nu}$.)\n",
    "\n",
    "and so doing the Taylor expansion,\n",
    "$$\n",
    "f(x \\pm \\nu h) =\\sum_{j=0}^{p} \\frac{ f^{(j)}(x) }{j!} ( \\pm \\nu h)^j + \\mathcal{O}(h^{p+1}) = \\sum_{j'=0}^{p'} \\frac{ f^{(2j')}(x) }{ (2j')! } (\\nu h)^{2j'} + \\sum_{j'=0}^{p'} \\frac{ f^{(2j'+1) }(x) }{ (2j'+1)!} (\\nu h)^{2j'+1} (\\pm 1) + \\mathcal{O}(h^{p+1} )\n",
    "$$\n",
    "where $p := 2p'+1$.  \n",
    "\n",
    "Clearly, for $f(x+\\nu h) + f(x-\\nu h)$, all the \"odd\" terms in the Taylor series expansion cancel each other out, leaving\n",
    "$$\n",
    "f(x+\\nu h) + f(x-\\nu h) -2f(x)= \\sum_{j'=1}^{p'} \\frac{ 2f^{(2j')}(x) }{(2j')!} (\\nu h)^{2j'} + O(h^{p+1}) =  \\frac{ 2f''(x)}{2} (\\nu h)^2 + \\sum_{j'=2}^{p'} \\frac{2(\\nu h)^{2j' } }{ (2j')! } f^{(2j') }(x) + \\mathcal{O}(h^{p+1} ) \n",
    "$$\n",
    "\n",
    "and so with our previous assumption (caveat),  \n",
    "\n",
    "$$\n",
    "\\sum_{\\nu =1}^{N} C_{\\nu} ( f(x+ \\nu h) + f(x-\\nu h) -2f(x)) \\frac{1}{h^2} = \\sum_{\\nu =1}^N C_{\\nu} \\left(  \\nu^2 f''(x)h^2 + \\sum_{j'=2}^{p'} \\frac{ 2\\nu ^{2j'} }{ (2j')! } f^{(2j') }(x) h^{2j'} \\right) \\frac{1}{h^2} + \\mathcal{O}(h^{p} ) = \\sum_{\\nu =1}^N \\left[ C_{\\nu} \\nu^2  f''(x) + C_{\\nu} \\sum_{j'=2}^{p'} \\frac{ 2\\nu^{2j'} }{ (2j')!} f^{(2j') }(x)h^{2(j'-1) } \\right] + \\mathcal{O}(h^p)\n",
    "$$\n",
    "\n",
    "Then we obtain a system of linear equations that we'd want to solve:\n",
    "\n",
    "$$\n",
    "\\sum_{\\nu =1}^N \\nu^2 C_{\\nu}  = 1 \n",
    "$$  \n",
    "\n",
    "$$\n",
    "\\sum_{j'=2}^{p'} \\left( \\sum_{\\nu =1}^N C_{\\nu} \\frac{ 2\\nu^{2j'} }{ (2j')! } \\right)= 0 \n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_C_nu_d2p(NU):\n",
    "    C_nu_dp = sympy.zeros(NU,NU+1)\n",
    "    C_nu_dp[:NU-1,:NU] = Matrix(NU-1,NU,lambda i,j: (j+1)**(2*(i+2)))\n",
    "\n",
    "    C_nu_dp[NU-1,:NU]  = Matrix(1,NU,lambda i,j:(j+1)**2)\n",
    "    C_nu_dp[NU-1,NU]   = Rat(1)\n",
    "\n",
    "    # sanity check\n",
    "    pprint( C_nu_dp)\n",
    "    Cnus = symbols(' '.join( ['C_'+str(nu)for nu in range(1,NU+1)]), real=True)\n",
    "    return solve_linear_system( C_nu_dp, *Cnus )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  16  0⎤\n",
      "⎢        ⎥\n",
      "⎣1  4   1⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAAyBAMAAAAegQJiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEPUvp2byvrASDcwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{4}{3}, \\quad C_{2} : - \\frac{1}{12}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 4/3, C₂: -1/12}"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_d2p(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  16  81   0⎤\n",
      "⎢             ⎥\n",
      "⎢1  64  729  0⎥\n",
      "⎢             ⎥\n",
      "⎣1  4    9   1⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASUAAAAyBAMAAADo7dDTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{3}{2}, \\quad C_{2} : - \\frac{3}{20}, \\quad C_{3} : \\frac{1}{90}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 3/2, C₂: -3/20, C₃: 1/90}"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_d2p(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  16    81    256   0⎤\n",
      "⎢                      ⎥\n",
      "⎢1  64   729   4096   0⎥\n",
      "⎢                      ⎥\n",
      "⎢1  256  6561  65536  0⎥\n",
      "⎢                      ⎥\n",
      "⎣1   4    9     16    1⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaEAAAAyBAMAAAAHA3moAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dIma7q80y\ndlRAyO8oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGRElEQVRoBd1ZXWhcRRQ+dzfZm2w2N0ulKbaU\nrBbRRkO2jRGSilkf+uRDGlCrVJqIkoJBsjRQER9yqRSrUNI+GGhejMRCiGiDiA9icbHQlz4kpYgo\nSFZKUUFMtFpNoq5zZubu3Nncn5mhD2kGMnNmzs93vr1z5+cGgJRteazv+vJ6waNQ92zRE+/qtmE1\ny/OfyIURsb7fUwjT3anxO4hx4kWWlL0amlwS4ONQZZjipvdLhRnI40YYIoQzIGR7jcnp38RYjfQT\nwBs1Q3Fda+aiHiMDDF8OO3v92feVqKpl3mchi3sBWuURhd6SHiMjDJFGs59Rv0sVbdNCXyP1n4OH\naobiu5qMjDBEFhKj8TxVRDBKV066wllR0mRkhCFSkRhxKhGMoHNdbwohkiYjIwxjRs4Ho08LZ0VJ\nk5ERhkhF8xm9A81/aj8kTUZGGMaMDgMslIW3mqTJyAhDZKL3jKwVgLQrvNUkPUZmGCITPUZAfr/G\novBWk/QYmWGITDQZPZWFM8JZUdJkZIQhUtFklJnRP6n2Dn7uCsB4yQRDRE1eWJ8VPXk/sr6dm7Vz\nQhsl7Z6ac49HGWzU6YQX3ppAMqOvXXC6XREsQmp+DuAG+dMpGuFFWF0gidG9eLfoU1oGrEmyRyX/\nEsAKkkZ4EU0biDNayGGML0mWcBCl2NKEx926Q7F2fgON8MJNG6jlLHWeKJAm+R/Kl7GKLX1lYpLJ\nx9r5DHTCCzdtoCb2Mw9jiPEhrHdghSU1TZvAyr6Nw1bRU77iCRGtRngRRR/I+Ru9E/SF6FxkkTLz\ntN3OGjZWUyf+ZQOJnn0oWEdq9EFdL3xHbxbVUeGFuwcEcBYHVYAuFonhAwU0Hy1hDZneIdpGVal/\nmPZB+DDKTNLx8Ek3SZOTdKEdDwgcfHGVSppcFXa9T00H6W+XBWfI59lSweIboGI9jW/BMVjAX4SV\nXcNYvgCgLivesNfy8Mlcg0/FwlOHQC8PCLb5GFmfIdBICYK9by5CK506MIrYTklm5OUjtyma1C/w\nMowXZE14zwsPWs+IA8E1H6NwDKq5sgjQXUKR7kTkQ5P0jKjNhortRHkyvpzdoAwZ8MLD9lKIRcCw\nB5T5QZlREi+oSXLAJmtdjlT3KTGyfyemCSTzKvlTK1745ktq9tTKA0pYyow6S+g5ghWuezdIlvwZ\nSYuRtY4Woiy74OBcTeBzClmC2uaFPZG88HzWSeGFYQjQW4yRwlpXx5bhpQLGvGdqBlvOSN6PhvGB\niGL1zD2KvTfZUOB+lMJDla/w8GDTnVwOL8wCgaxF/owCgYQzkRpXaJedgrhGfo+armWeIAqnyLVS\nU5erK0kDrHMpe7lMpB8DVOQtsukeyHX2/XvRlJQoIKer65ZLreSK+SQ6HgFr/8kS1fGPqW15n6Vz\nyNeB+kolR/oJ/1hVvvrrz9lqRwiDFXzPICdGhJQqNg+IHozR2YhfriOBwA58j6iP9SkcgKaydYyG\nlc7eDMh+bG3RB9nU/g32dvuGhDi4catC5ScfFUlt+cNUfTId7aj0ylIZbgP9ch0JBG+vlT0XX0t9\nGvNwHk4BPEwVAYx8DiimWd9ljVo9Tc0aVIwPFnAtwHu1CRD16cdfaMrbSJQZqWRXtZmuSgoCroE+\nRgoewoQyehf7awD9ZRQUGJ2+XkJLnTJ75nF189YcZ2QAlEaf9a+uF6xbhBGd5fGMGrP2H+rpMctz\nGoej4y9lGSMTIPSx1l04ijsxW7DjGZEcJ3UZkV1hSNknQU4t/PuUARBMWpUsdO8gz0iDkdqnB4mC\nzieIZ7IeIwMgcholt8+JJzVmHbkAdRakbGM76XlIqM7U1wDeK1NGBkB4O+ssjBBGLlkZlsuYWfys\nI9fTPjLTdUr9PCQDN8SAIGTKcEYGQEB9LhBGhfOkomnGM8rzc2xANmFD5EaSmg9T1ow/T+7ORfqM\nDICA+iyQ9wh32AM0dDyjMWg4XJNFXNfOw6linBHXnwaHHJZxZTAAYj7pnHUUUmXrBRqSH+g4saAs\n7P0z2aDxqLGx9n1Rar+urqunAPTLtQkQ89nTWwKr47sSjcv/s0xm/lYp/BrRqPoib37e7AoLGbKc\nb41irXIeS4tbgxCcGOBEGta0X/9N+RM4R6o8dk5vygx1k7rqosf/XcHG2nFcLkIAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : \\frac{8}{5}, \\quad C_{2} : - \\frac{1}{5}, \\quad C_{3} : \\frac{8}{315}, \\quad C_{4} : - \\frac{1}{560}\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 8/5, C₂: -1/5, C₃: 8/315, C₄: -1/560}"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_C_nu_d2p(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAw4AAAA4CAMAAABjV19SAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{6} + h^{5} x + h^{4} x^{2} + h^{3} x^{3} + h^{2} x^{4} + h x^{5} + x^{6}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                             \n",
       " 2 ⎜ d         ⎟│         ⎛ 6    5      4  2    3  3    2  4      5    6      \n",
       "h ⋅⎜────(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h⋅x  + x ; (h, \n",
       "   ⎜   2       ⎟│                                                             \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀                                                        \n",
       "\n",
       "            \n",
       "           ⎞\n",
       "x) → (0, 0)⎠\n",
       "            \n",
       "            "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 6\n",
    "( Rat(4,3)*(f(x+h).series(x+h,x0=x_0,n=p) + f(x-h).series(x-h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(-1,12)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)+f(x-2*h).series(x-2*h,x0=x_0,n=p) -2*f(x_0))).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{8} + h^{7} x + h^{6} x^{2} + h^{5} x^{3} + h^{4} x^{4} + h^{3} x^{5} + h^{2} x^{6} + h x^{7} + x^{8}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                             \n",
       " 2 ⎜ d         ⎟│         ⎛ 8    7      6  2    5  3    4  4    3  5    2  6  \n",
       "h ⋅⎜────(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  +\n",
       "   ⎜   2       ⎟│                                                             \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀                                                        \n",
       "\n",
       "                            \n",
       "    7    8                 ⎞\n",
       " h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "                            \n",
       "                            "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 8\n",
    "( Rat(3,2)*(f(x+h).series(x+h,x0=x_0,n=p) + f(x-h).series(x-h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(-3,20)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)+f(x-2*h).series(x-2*h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(1,90)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)+f(x-3*h).series(x-3*h,x0=x_0,n=p) -2*f(x_0)) ).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ApzOQqTMnzPbV6kTka2qciulMz8W9MLlyO0v5wwJO2KuifKm3qKJ9uidtkmjCgDCwz0Dc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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{10} + h^{9} x + h^{8} x^{2} + h^{7} x^{3} + h^{6} x^{4} + h^{5} x^{5} + h^{4} x^{6} + h^{3} x^{7} + h^{2} x^{8} + h x^{9} + x^{10}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                             \n",
       " 2 ⎜ d         ⎟│         ⎛ 10    9      8  2    7  3    6  4    5  5    4  6 \n",
       "h ⋅⎜────(f(ξ₁))⎟│      + O⎝h   + h ⋅x + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  + h ⋅x  \n",
       "   ⎜   2       ⎟│                                                             \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀                                                        \n",
       "\n",
       "                                              \n",
       "   3  7    2  8      9    10                 ⎞\n",
       "+ h ⋅x  + h ⋅x  + h⋅x  + x  ; (h, x) → (0, 0)⎠\n",
       "                                              \n",
       "                                              "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 10\n",
    "( Rat(8,5)*(f(x+h).series(x+h,x0=x_0,n=p) + f(x-h).series(x-h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(-1,5)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)+f(x-2*h).series(x-2*h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    "Rat(8,315)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)+f(x-3*h).series(x-3*h,x0=x_0,n=p) -2*f(x_0)) + \\\n",
    " Rat(-1,560)*(f(x+4*h).series(x+4*h,x0=x_0,n=p)+f(x-4*h).series(x-4*h,x0=x_0,n=p) -2*f(x_0))\n",
    ").subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# finite difference sanity check on Python numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib\n",
    "import pylab\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = {'svg','png'}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "l_x = 2.* np.pi\n",
    "l_y = 2.* np.pi\n",
    "l_z = 2.* np.pi\n",
    "L_x = 64 \n",
    "L_y = 64 \n",
    "L_z = 64\n",
    "f_grid3d = np.zeros((L_x,L_y,L_z))\n",
    "df_grid3d_result = np.zeros((L_x,L_y,L_z))\n",
    "h_x = l_x/L_x; h_y = l_y/L_y; h_z = l_z/L_z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for i_z in range(L_z):\n",
    "    for i_y in range(L_y):\n",
    "        for i_x in range(L_x):\n",
    "            f_grid3d[i_x][i_y][i_z] = np.sin( i_x*h_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(64,)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_grid3d[:,0,0].shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe5ad8a5590>]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"370.2828125\" xlink:href=\"#m3c14f77237\" y=\"67.8390625\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.5 -->\n",
       "      <g transform=\"translate(15.5796875 70.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_26\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"35.4828125\" xlink:href=\"#m4383fd3d8c\" y=\"12.0390625\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_27\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"370.2828125\" xlink:href=\"#m3c14f77237\" y=\"12.0390625\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(15.5796875 14.7984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pccdd73348c\">\n",
       "   <rect height=\"223.2\" width=\"334.8\" x=\"35.4828125\" y=\"12.0390625\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe5b7003310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.plot( f_grid3d[:,5,5] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "RAD = 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for i_z in range(L_z-RAD):\n",
    "    for i_y in range(L_y-RAD):\n",
    "        for i_x in range(L_x-RAD):\n",
    "#            if (i_z + RAD >= L_z): \n",
    "            df_grid3d_result[i_x][i_y][i_z] = \\\n",
    "                (3./(4.*h_x))*(f_grid3d[i_x+1][i_y][i_z]-f_grid3d[i_x-1][i_y][i_z]) + \\\n",
    "                (-3./(20.*h_x))*(f_grid3d[i_x+2][i_y][i_z]-f_grid3d[i_x-2][i_y][i_z]) + \\\n",
    "                (1./(60.*h_x))*(f_grid3d[i_x+3][i_y][i_z]-f_grid3d[i_x-3][i_y][i_z]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe5ad7de150>]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       " <defs>\n",
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe5ad952dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.plot( df_grid3d_result[:,5,5] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe5ad6c9610>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe5deb0ff90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.plot( df_grid3d_result[:,0,0] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.19509032201612872"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_grid3d[-2][0][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p = 6\n",
    "( Rat(3,4)*(f(x+h).series(x+h,x0=x_0,n=p) - f(x-h).series(x-h,x0=x_0,n=p)) + \\\n",
    "Rat(-3,20)*(f(x+2*h).series(x+2*h,x0=x_0,n=p)-f(x-2*h).series(x-2*h,x0=x_0,n=p)) + \\\n",
    "Rat(1,60)*(f(x+3*h).series(x+3*h,x0=x_0,n=p)-f(x-3*h).series(x-3*h,x0=x_0,n=p))).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Instantons\n",
    "\n",
    "cf. Valery Rubakov. **Classical Theory of Gauge Fields**.  Princeton, 2002.  Ch. 7.  7.1 Kink"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t = Symbol('t',real=True)\n",
    "phi1pl1 = Symbol(r'\\varphi')# phi1pl1 denote phi 1+1 spacetime dimension; pl for \"plus\"\n",
    "phi1pl1 = Function(r'\\varphi')(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAUBAMAAADbzbjtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVDJEzbsQiWarmSJ2\n3e+oEHSfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSElEQVQoFW2SMUvDUBDHf2nSJG3a4uSggxn8\nAMVNEAyCm6CT6GQ2EUQ7iYNgwEWUQiYHdejgB+joUKFTQXEoKDr4JaKugt5LXpoM3pD7/+9/73J3\n70Fu9znQ/rLEvY0SUbDWLwJWH/ujoGAHBbsAIymooJOCyal6UFBBBxPmjaHZmVAFrDaGj7m+SaPH\n4s72UGLyt+ebJanp9HBg/uyaagRzqQS7dIev4AbIODE1mlNwp8UjTsNZ6W9MDzekTqsN71oMWVWo\nktChAQMl2t9ahB8txizAlSprJoSZXPlS3hxzLIX3fNWQExi+xGC6kTBKG7LWVmZe0r7r8T6HgZz4\nrCZeJDuRsbu/qhGp4W1FNN9kzNvR07mELF8SJElML11GyO1BQMtPmSxeWWmDavHLaZBqX3l7KJ/M\n0it7zHB22W4uobM1/+eZ/AH5sEEdbJqrCAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\varphi{\\left (t,x \\right )}$$"
      ],
      "text/plain": [
       "\\varphi(t, x)"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi1pl1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\varphi{\\left (t,x_{0} \\right )} + \\left(x - x_{0}\\right) \\left. \\frac{\\partial}{\\partial \\xi_{2}} \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }} + \\frac{1}{2} \\left(x - x_{0}\\right)^{2} \\left. \\frac{\\partial^{2}}{\\partial \\xi_{2}^{2}}  \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }} + \\mathcal{O}\\left(\\left(x - x_{0}\\right)^{3}; x\\rightarrowx_{0}\\right)$$"
      ],
      "text/plain": [
       "                                                                  ⎛  2        \n",
       "                                                                2 ⎜ ∂         \n",
       "                                                        (x - x₀) ⋅⎜────(\\varph\n",
       "                                                                  ⎜   2       \n",
       "                          ⎛ ∂                 ⎞│                  ⎝∂ξ₂        \n",
       "\\varphi(t, x₀) + (x - x₀)⋅⎜───(\\varphi(t, ξ₂))⎟│      + ──────────────────────\n",
       "                          ⎝∂ξ₂                ⎠│ξ₂=x₀                     2   \n",
       "\n",
       "         ⎞│                            \n",
       "         ⎟│                            \n",
       "i(t, ξ₂))⎟│                            \n",
       "         ⎟│                            \n",
       "         ⎠│ξ₂=x₀    ⎛        3        ⎞\n",
       "──────────────── + O⎝(x - x₀) ; x → x₀⎠\n",
       "                                       "
      ]
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi1pl1.series(x=x,x0=x_0,n=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left. \\frac{\\partial}{\\partial x} \\varphi{\\left (t,x \\right )} \\right|_{\\substack{ x=x + x_{0} }}^{2} + \\frac{\\partial}{\\partial t} \\varphi{\\left (t,x_{0} \\right )}^{2} + 2 \\frac{\\partial}{\\partial t}\\left(\\frac{1}{2} \\left(x - x_{0}\\right)^{2} \\left. \\frac{\\partial^{2}}{\\partial \\xi_{2}^{2}}  \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }}\\right) \\frac{\\partial}{\\partial t} \\varphi{\\left (t,x_{0} \\right )} + 2 \\frac{\\partial}{\\partial t}\\left(\\left(x - x_{0}\\right) \\left. \\frac{\\partial}{\\partial \\xi_{2}} \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }}\\right) \\frac{\\partial}{\\partial t} \\varphi{\\left (t,x_{0} \\right )} + \\frac{\\partial}{\\partial t}\\left(\\left(x - x_{0}\\right) \\left. \\frac{\\partial}{\\partial \\xi_{2}} \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }}\\right)^{2} + \\mathcal{O}\\left(\\left(x - x_{0}\\right)^{3}; x\\rightarrowx_{0}\\right)$$"
      ],
      "text/plain": [
       "                                                            ⎛          ⎛  2   \n",
       "                                                            ⎜        2 ⎜ ∂    \n",
       "                                                            ⎜(x - x₀) ⋅⎜────(\\\n",
       "                            2                       2       ⎜          ⎜   2  \n",
       "⎛∂                ⎞│            ⎛∂                 ⎞      ∂ ⎜          ⎝∂ξ₂   \n",
       "⎜──(\\varphi(t, x))⎟│          + ⎜──(\\varphi(t, x₀))⎟  + 2⋅──⎜─────────────────\n",
       "⎝∂x               ⎠│x=x + x₀    ⎝∂t                ⎠      ∂t⎝                 \n",
       "\n",
       "              ⎞│     ⎞                                                        \n",
       "              ⎟│     ⎟                                                        \n",
       "varphi(t, ξ₂))⎟│     ⎟                                                        \n",
       "              ⎟│     ⎟                                                        \n",
       "              ⎠│ξ₂=x₀⎟ ∂                      ∂ ⎛         ⎛ ∂                 \n",
       "─────────────────────⎟⋅──(\\varphi(t, x₀)) + 2⋅──⎜(x - x₀)⋅⎜───(\\varphi(t, ξ₂))\n",
       " 2                   ⎠ ∂t                     ∂t⎝         ⎝∂ξ₂                \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                        2     \n",
       "⎞│     ⎞ ∂                    ⎛∂ ⎛         ⎛ ∂                 ⎞│     ⎞⎞     ⎛\n",
       "⎟│     ⎟⋅──(\\varphi(t, x₀)) + ⎜──⎜(x - x₀)⋅⎜───(\\varphi(t, ξ₂))⎟│     ⎟⎟  + O⎝\n",
       "⎠│ξ₂=x₀⎠ ∂t                   ⎝∂t⎝         ⎝∂ξ₂                ⎠│ξ₂=x₀⎠⎠      \n",
       "\n",
       "                  \n",
       "                  \n",
       "                  \n",
       "                  \n",
       "        3        ⎞\n",
       "(x - x₀) ; x → x₀⎠\n",
       "                  "
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((phi1pl1.diff(t)*phi1pl1.diff(t)) + (phi1pl1.diff(x)*phi1pl1.diff(x))).series(x=x,x0=x_0,n=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAAtBAMAAADFBk2rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEGYiRN27ds0yie+r\nmVS4Q9qgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADhUlEQVRIDbVVTUgUYRh+Zt3Zn9mfBjoEFrhk\nQofCvVS33EtEIWlpZD/EaFfD9dShhI2wNojyUggdnOygF0moY9RGhzoY6aGILhlBUaBuPxCBuL3f\nz3wz07oEMvPCvu/zPt87z77z/Q2gtSIcay+Eo6tb4ejGv4ej2/Q7HF2MBq07c95kku1jwQob+XSJ\nKbbYzAdnej75k6lNzAanyZT0QtMvCskB5gO1LNtit83VBqLG24V1RxrxqlgvAtGH+CCJZEGNcHAf\nL/2EzBrxNEyCZDP06ypgYpElwB0RlO/BRF4lHtCIp5JOVhaZp3Z/0ERbLAMui6B8LzoKKvGARjyV\nGDlyh3EIMeo8y7cFMuIlPAJYMb2ZixnflHNzjvjFeJQ2ws3lfu9QR73ICe+4BzN+ryfnMJ2nMAKk\narWqd4zNtt+MeX/uZJy/Amg9DsOiViLXwZDfnvhTynY7zBcHiMh4tjhZX1+YIlrPiRLXJ+vuy6gd\nHRPjfl3O63kgU3IfJ7SVfhnLR1ESmf2XGV/+bArOr8v5PTSi2+RcSxSAeJ2K98+1kaJ2DE9rNfmQ\n1M2OIXqrX/C0QJ1Dg0VeoF14fpAmgU17tO6tY5YUoWBkZ3HGTSF1DWB8xyvBD1BYkSXNaCv20RuX\naJP9kZwKiYqC2J/pxjc3dXRph1aQFvxZCu9kyT08MD/RMnbTrq67xHR6C230OFnfmJkq4oijO10u\nny6Xb1A6j4iJjOAfUTglS0xcZShZJd01wdW4EQFvv+gy5RUiquQ82IgD+wRFuprbG/9Gcl2XE3V+\n3TeALXkWpG6FDj6eCZ7mIVqFKRIxq9Fu6lncCYLmPm15kkHQ2rsmdSdpJrZJntbNKMmij/EqhsW6\n1e8H386bRrOrqvpNXLu+1Cv5GToDlS1oKVHfa7Equ4Uy9nqnwPdP6Umqc032i7aas1Q4QGs0kINO\nF5j2YnhkOxUn6FXqzxtoJRqZo5vJqQp2jpnR/nKMHcFE0cmQnBfwvWLqwGbJpKgjaezeYWZzz90U\n+RZT5c7n4q5iGoIuz8g5jrWiorQSwV0qxUVL4PSiyzVAlzy8uNcjLhPLEz6p8qadlsD8/xT9X8B2\ngM9aKYvYijISlsRLitsw8HycWpVutrBhPefBBQdAKypdxQUBDISj+zUcXc0ORzc7N/f6sbu3g5hZ\nRyNlOSjYuCkcXWNoNRdsoyGq/QV0A8J6yEc3HAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\lambda}{4} \\left(- v^{2} + \\varphi^{2}{\\left (t,x \\right )}\\right)^{2}$$"
      ],
      "text/plain": [
       "                               2\n",
       "        ⎛   2          2      ⎞ \n",
       "\\lambda⋅⎝- v  + \\varphi (t, x)⎠ \n",
       "────────────────────────────────\n",
       "               4                "
      ]
     },
     "execution_count": 199,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lambdavariablesymbol = Symbol(r'\\lambda')\n",
    "v = Symbol('v')\n",
    "mu = Symbol(r'\\mu')\n",
    "m = Symbol('m')\n",
    "lambdavariablesymbol/Rat(4)*(phi1pl1*phi1pl1-v**2)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\mu^{4}}{4 \\lambda} + \\left(x - x_{0}\\right) \\left(\\lambda \\varphi^{3}{\\left (t,x_{0} \\right )} \\left. \\frac{\\partial}{\\partial \\xi_{2}} \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }} - \\mu^{2} \\varphi{\\left (t,x_{0} \\right )} \\left. \\frac{\\partial}{\\partial \\xi_{2}} \\varphi{\\left (t,\\xi_{2} \\right )} \\right|_{\\substack{ \\xi_{2}=x_{0} }}\\right) - \\frac{\\mu^{2}}{2} \\varphi^{2}{\\left (t,x_{0} \\right )} + \\frac{\\lambda}{4} \\varphi^{4}{\\left (t,x_{0} \\right )} + \\mathcal{O}\\left(\\left(x - x_{0}\\right)^{2}; x\\rightarrowx_{0}\\right)$$"
      ],
      "text/plain": [
       "      4                                                                       \n",
       "   \\mu               ⎛               3        ⎛ ∂                 ⎞│          \n",
       "───────── + (x - x₀)⋅⎜\\lambda⋅\\varphi (t, x₀)⋅⎜───(\\varphi(t, ξ₂))⎟│      - \\m\n",
       "4⋅\\lambda            ⎝                        ⎝∂ξ₂                ⎠│ξ₂=x₀     \n",
       "\n",
       "                                                    2        2                \n",
       " 2                ⎛ ∂                 ⎞│     ⎞   \\mu ⋅\\varphi (t, x₀)   \\lambd\n",
       "u ⋅\\varphi(t, x₀)⋅⎜───(\\varphi(t, ξ₂))⎟│     ⎟ - ──────────────────── + ──────\n",
       "                  ⎝∂ξ₂                ⎠│ξ₂=x₀⎠            2                   \n",
       "\n",
       "         4                              \n",
       "a⋅\\varphi (t, x₀)    ⎛        2        ⎞\n",
       "───────────────── + O⎝(x - x₀) ; x → x₀⎠\n",
       "     4                                  "
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(lambdavariablesymbol/Rat(4)*(phi1pl1*phi1pl1-v**2)**2).series(x=x,x0=x_0,n=2).subs(v, \\\n",
    "                                                        mu/sympy.sqrt( lambdavariablesymbol))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Dealing with Boundary conditions (with Finite Difference)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I found that I needed to represent boundary conditions that are \"discretized\" i.e. \"numerical boundary condition\" - for example on pp. 148 of Cebeci, Shao, Kafyeke, Laurendeau, (E5.2.3), I need to figure out if this is a derivative or second-order derivative:\n",
    "\n",
    "$u_I^n = 2u^n_{I-1} - u^n_{I-2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                                 \n",
       " 2 ⎜ d         ⎟│         ⎛ 3    2        2    3                 ⎞\n",
       "h ⋅⎜────(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "   ⎜   2       ⎟│                                                 \n",
       "   ⎝dξ₁        ⎠│ξ₁=x₀                                            "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(f(x).series(x,x0=x_0,n=3)- 2 * f(x-h).series(x-h,x0=x_0,n=3) + \\\n",
    " f(x-2*h).series(x-2*h,x0=x_0,n=3)).subs(x-x_0,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So indeed this is essentially\n",
    "\n",
    "$ \\frac{ \\partial^2 u}{ \\partial x^2} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Also consider pp. 338 of Cebeci, Shao, Kafyeke, Laurendeau (2005), namely (11.5.7a), (11.5.7b), which purports the second-order accuracy of a boundary condition, involving a first-order partial derivative:\n",
    "\n",
    "$ \\frac{ \\partial u^x }{ \\partial y } = 0 $ and \n",
    "\n",
    "$\\frac{\\partial p }{ \\partial y} = 0 $\n",
    "\n",
    "cf. on the \"symmetry plane\" of $y=H$ (these boundary conditions, $\\lbrace (x,y) \\in \\mathbb{R}^2 | y=H \\rbrace$ is a *submanifold* of $\\mathbb{R}^2$ as it's a level set or from the locally imbedded theorem (from differential topology))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkIAAAA3CAMAAADXCdVpAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "  ⎛ d        ⎞│         ⎛ 3    2        2    3                 ⎞\n",
       "h⋅⎜───(f(ξ₁))⎟│      + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "  ⎝dξ₁       ⎠│ξ₁=x₀                                            "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p=3\n",
    "( Rat(3,2)*f(x).series(x,x0=x_0,n=p).subs(x-x_0,0) + \\\n",
    "  Rat(-2)*( f(x-h).series(x-h,x0=x_0,n=p) ).subs(x-x_0,0) + \\\n",
    "Rat(1,2)*( f(x-2*h).series(x-2*h,x0=x_0,n=p)).subs(x-x_0,0) )\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\frac{2 h}{3} \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=x_{0} }} + \\mathcal{O}\\left(h^{3} + h^{2} x + h x^{2} + x^{3}; \\left ( h, \\quad x\\right )\\rightarrow\\left ( 0, \\quad 0\\right )\\right)$$"
      ],
      "text/plain": [
       "    ⎛ d        ⎞│                                                 \n",
       "2⋅h⋅⎜───(f(ξ₁))⎟│                                                 \n",
       "    ⎝dξ₁       ⎠│ξ₁=x₀    ⎛ 3    2        2    3                 ⎞\n",
       "────────────────────── + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠\n",
       "          3                                                       "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "( Rat(1)*f(x).series(x,x0=x_0,n=p).subs(x-x_0,0) + \\\n",
    "  Rat(-4,3)*( f(x-h).series(x-h,x0=x_0,n=p) ).subs(x-x_0,0) + \\\n",
    "Rat(1,3)*( f(x-2*h).series(x-2*h,x0=x_0,n=p)).subs(x-x_0,0) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then \"set this\" first order partial derivative to zero and then you obtain the \"scheme\" to \"discretize\" the boundary condition.  \n",
    "\n",
    "$u_{i,J} = \\frac{ 4 u_{i,J-1} - u_{i,J-2} }{ 3 } $\n",
    "\n",
    "$p_{i,J} = \\frac{ 4p_{i,J-1} -p_{i,J-2} }{ 3 } $\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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