Wavefunctions/Explain_Bspline.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from collections import defaultdict\n",
    "from IPython.display import display\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# B-Splines in QMC\n",
    "\n",
    "Bsplines are used in QMC as a compact basis for single-particle orbitals, and are also used for Jastrow factors.\n",
    "\n",
    "The spline routines in QMCPACK were also packaged into a library named 'einspline', so the code could be reused in other QMC codes.\n",
    "\n",
    "This notebook is focused on deriving the evaluation code and coefficient solvers contained in einspline.\n",
    "\n",
    "For more information see\n",
    "* Einspline documentation page http://einspline.sourceforge.net/background.shtml\n",
    "* The paper that introduce b-splines to QMC: 'An efficient localized basis set for quantum Monte Carlo calculations on condensed matter' https://arxiv.org/abs/cond-mat/0407037\n",
    "* Comparison of some different spline types: 'Comparison of polynomial approximations to speed up planewave-based quantum Monte Carlo calculations'  https://arxiv.org/abs/1309.6250\n",
    "\n",
    "\n",
    "### B-spline basis\n",
    "To start, we will use the the bspline code built into Sympy.  See the wikipedia page for more about B-splines: https://en.wikipedia.org/wiki/B-spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\begin{cases} \\frac{x^{3}}{6} & \\text{for}\\: x \\geq 0 \\wedge x < 1 \\\\- \\frac{x^{3}}{2} + 2 x^{2} - 2 x + \\frac{2}{3} & \\text{for}\\: x \\geq 1 \\wedge x < 2 \\\\\\frac{x^{3}}{2} - 4 x^{2} + 10 x - \\frac{22}{3} & \\text{for}\\: x \\geq 2 \\wedge x < 3 \\\\- \\frac{x^{3}}{6} + 2 x^{2} - 8 x + \\frac{32}{3} & \\text{for}\\: x \\geq 3 \\wedge x \\leq 4 \\\\0 & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧           3                             \n",
       "⎪          x                              \n",
       "⎪          ──            for x ≥ 0 ∧ x < 1\n",
       "⎪          6                              \n",
       "⎪                                         \n",
       "⎪   3                                     \n",
       "⎪  x       2         2                    \n",
       "⎪- ── + 2⋅x  - 2⋅x + ─   for x ≥ 1 ∧ x < 2\n",
       "⎪  2                 3                    \n",
       "⎪                                         \n",
       "⎨ 3                                       \n",
       "⎪x       2          22                    \n",
       "⎪── - 4⋅x  + 10⋅x - ──   for x ≥ 2 ∧ x < 3\n",
       "⎪2                  3                     \n",
       "⎪                                         \n",
       "⎪   3                                     \n",
       "⎪  x       2         32                   \n",
       "⎪- ── + 2⋅x  - 8⋅x + ──  for x ≥ 3 ∧ x ≤ 4\n",
       "⎪  6                 3                    \n",
       "⎪                                         \n",
       "⎩          0                 otherwise    "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = Symbol('x')\n",
    "knots = [0,1,2,3,4,5,6]\n",
    "\n",
    "# Third-order bspline\n",
    "sym_basis = bspline_basis_set(3, knots, xs)\n",
    "# Form for one basis function\n",
    "sym_basis[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fedaf5b1910>,\n",
       " <matplotlib.lines.Line2D at 0x7fedaf5b1b10>,\n",
       " <matplotlib.lines.Line2D at 0x7fedaf5c4210>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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LyiX7coUO4ezXjnM7KKouYlb4rF9+0+AC/RdBxh7IS7Z5NkclhGB2xGzi8+NJ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\nzt0wDCMFMeduGIaRgphzNwzDSEH+D8/8rwA7m9zhAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fedb19b2ed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot some basis functions\n",
    "nbasis_to_plot = 3\n",
    "npoints_to_plot = 40\n",
    "basis_y = np.zeros((nbasis_to_plot, npoints_to_plot))\n",
    "xvals = np.zeros(npoints_to_plot)\n",
    "for i in range(npoints_to_plot):\n",
    "    xv = i*.1 + 1.0\n",
    "    for j in range(3):\n",
    "        basis_y[j,i] = sym_basis[j].subs(xs,xv)\n",
    "    xvals[i] = xv\n",
    "plt.plot(xvals, basis_y[0,:], xvals, basis_y[1,:], xvals, basis_y[2,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function approximation\n",
    "\n",
    "### For fixed (non-periodic) boundary conditions\n",
    "To fully represent an interval (say, 0.0 - 5.0, with a knot spacing of 1.0), we need parts of basis functions outside the interval as well.  To approximate a function, we evaluate at the M knots (there are 6 knots for this example).  There will be M+2 basis functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Number of basis functions = ', 8)\n"
     ]
    },
    {
     "data": {
      "image/png": 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MG17vl/SZkK4Qen8EasdDJtFnX+Ylh9lCGxO7xnBOi9eV5gnf7x3pcwIGDVhF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\n",
      "text/latex": [
       "$$\\left [ \\begin{cases} \\frac{x^{3}}{6} + \\frac{3 x^{2}}{2} + \\frac{9 x}{2} + \\frac{9}{2} & \\text{for}\\: x \\geq -3 \\wedge x < -2 \\\\- \\frac{x^{3}}{2} - \\frac{5 x^{2}}{2} - \\frac{7 x}{2} - \\frac{5}{6} & \\text{for}\\: x \\geq -2 \\wedge x < -1 \\\\\\frac{x^{3}}{2} + \\frac{x^{2}}{2} - \\frac{x}{2} + \\frac{1}{6} & \\text{for}\\: x \\geq -1 \\wedge x < 0 \\\\- \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - \\frac{x}{2} + \\frac{1}{6} & \\text{for}\\: x \\geq 0 \\wedge x \\leq 1 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} + x^{2} + 2 x + \\frac{4}{3} & \\text{for}\\: x \\geq -2 \\wedge x < -1 \\\\- \\frac{x^{3}}{2} - x^{2} + \\frac{2}{3} & \\text{for}\\: x \\geq -1 \\wedge x < 0 \\\\\\frac{x^{3}}{2} - x^{2} + \\frac{2}{3} & \\text{for}\\: x \\geq 0 \\wedge x < 1 \\\\- \\frac{x^{3}}{6} + x^{2} - 2 x + \\frac{4}{3} & \\text{for}\\: x \\geq 1 \\wedge x \\leq 2 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} + \\frac{x^{2}}{2} + \\frac{x}{2} + \\frac{1}{6} & \\text{for}\\: x \\geq -1 \\wedge x < 0 \\\\- \\frac{x^{3}}{2} + \\frac{x^{2}}{2} + \\frac{x}{2} + \\frac{1}{6} & \\text{for}\\: x \\geq 0 \\wedge x < 1 \\\\\\frac{x^{3}}{2} - \\frac{5 x^{2}}{2} + \\frac{7 x}{2} - \\frac{5}{6} & \\text{for}\\: x \\geq 1 \\wedge x < 2 \\\\- \\frac{x^{3}}{6} + \\frac{3 x^{2}}{2} - \\frac{9 x}{2} + \\frac{9}{2} & \\text{for}\\: x \\geq 2 \\wedge x \\leq 3 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} & \\text{for}\\: x \\geq 0 \\wedge x < 1 \\\\- \\frac{x^{3}}{2} + 2 x^{2} - 2 x + \\frac{2}{3} & \\text{for}\\: x \\geq 1 \\wedge x < 2 \\\\\\frac{x^{3}}{2} - 4 x^{2} + 10 x - \\frac{22}{3} & \\text{for}\\: x \\geq 2 \\wedge x < 3 \\\\- \\frac{x^{3}}{6} + 2 x^{2} - 8 x + \\frac{32}{3} & \\text{for}\\: x \\geq 3 \\wedge x \\leq 4 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} - \\frac{x^{2}}{2} + \\frac{x}{2} - \\frac{1}{6} & \\text{for}\\: x \\geq 1 \\wedge x < 2 \\\\- \\frac{x^{3}}{2} + \\frac{7 x^{2}}{2} - \\frac{15 x}{2} + \\frac{31}{6} & \\text{for}\\: x \\geq 2 \\wedge x < 3 \\\\\\frac{x^{3}}{2} - \\frac{11 x^{2}}{2} + \\frac{39 x}{2} - \\frac{131}{6} & \\text{for}\\: x \\geq 3 \\wedge x < 4 \\\\- \\frac{x^{3}}{6} + \\frac{5 x^{2}}{2} - \\frac{25 x}{2} + \\frac{125}{6} & \\text{for}\\: x \\geq 4 \\wedge x \\leq 5 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} - x^{2} + 2 x - \\frac{4}{3} & \\text{for}\\: x \\geq 2 \\wedge x < 3 \\\\- \\frac{x^{3}}{2} + 5 x^{2} - 16 x + \\frac{50}{3} & \\text{for}\\: x \\geq 3 \\wedge x < 4 \\\\\\frac{x^{3}}{2} - 7 x^{2} + 32 x - \\frac{142}{3} & \\text{for}\\: x \\geq 4 \\wedge x < 5 \\\\- \\frac{x^{3}}{6} + 3 x^{2} - 18 x + 36 & \\text{for}\\: x \\geq 5 \\wedge x \\leq 6 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} - \\frac{3 x^{2}}{2} + \\frac{9 x}{2} - \\frac{9}{2} & \\text{for}\\: x \\geq 3 \\wedge x < 4 \\\\- \\frac{x^{3}}{2} + \\frac{13 x^{2}}{2} - \\frac{55 x}{2} + \\frac{229}{6} & \\text{for}\\: x \\geq 4 \\wedge x < 5 \\\\\\frac{x^{3}}{2} - \\frac{17 x^{2}}{2} + \\frac{95 x}{2} - \\frac{521}{6} & \\text{for}\\: x \\geq 5 \\wedge x < 6 \\\\- \\frac{x^{3}}{6} + \\frac{7 x^{2}}{2} - \\frac{49 x}{2} + \\frac{343}{6} & \\text{for}\\: x \\geq 6 \\wedge x \\leq 7 \\\\0 & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{x^{3}}{6} - 2 x^{2} + 8 x - \\frac{32}{3} & \\text{for}\\: x \\geq 4 \\wedge x < 5 \\\\- \\frac{x^{3}}{2} + 8 x^{2} - 42 x + \\frac{218}{3} & \\text{for}\\: x \\geq 5 \\wedge x < 6 \\\\\\frac{x^{3}}{2} - 10 x^{2} + 66 x - \\frac{430}{3} & \\text{for}\\: x \\geq 6 \\wedge x < 7 \\\\- \\frac{x^{3}}{6} + 4 x^{2} - 32 x + \\frac{256}{3} & \\text{for}\\: x \\geq 7 \\wedge x \\leq 8 \\\\0 & \\text{otherwise} \\end{cases}\\right ]$$"
      ],
      "text/plain": [
       "⎡⎧  3      2                                  ⎧  3                            \n",
       "⎢⎪ x    3⋅x    9⋅x   9                        ⎪ x     2         4             \n",
       "⎢⎪ ── + ──── + ─── + ─   for x ≥ -3 ∧ x < -2  ⎪ ── + x  + 2⋅x + ─   for x ≥ -2\n",
       "⎢⎪ 6     2      2    2                        ⎪ 6               3             \n",
       "⎢⎪                                            ⎪                               \n",
       "⎢⎪   3      2                                 ⎪      3                        \n",
       "⎢⎪  x    5⋅x    7⋅x   5                       ⎪     x     2   2               \n",
       "⎢⎪- ── - ──── - ─── - ─  for x ≥ -2 ∧ x < -1  ⎪   - ── - x  + ─     for x ≥ -1\n",
       "⎢⎪  2     2      2    6                       ⎪     2         3               \n",
       "⎢⎪                                            ⎪                               \n",
       "⎢⎨    3    2                                , ⎨     3                         \n",
       "⎢⎪   x    x    x   1                          ⎪    x     2   2                \n",
       "⎢⎪   ── + ── - ─ + ─     for x ≥ -1 ∧ x < 0   ⎪    ── - x  + ─       for x ≥ 0\n",
       "⎢⎪   2    2    2   6                          ⎪    2         3                \n",
       "⎢⎪                                            ⎪                               \n",
       "⎢⎪     3    2                                 ⎪   3                           \n",
       "⎢⎪    x    x    x   1                         ⎪  x     2         4            \n",
       "⎢⎪  - ── + ── - ─ + ─     for x ≥ 0 ∧ x ≤ 1   ⎪- ── + x  - 2⋅x + ─   for x ≥ 1\n",
       "⎢⎪    6    2    2   6                         ⎪  6               3            \n",
       "⎢⎪                                            ⎪                               \n",
       "⎣⎩          0                 otherwise       ⎩         0                other\n",
       "\n",
       "           ⎧    3    2                                 ⎧           3          \n",
       "           ⎪   x    x    x   1                         ⎪          x           \n",
       " ∧ x < -1  ⎪   ── + ── + ─ + ─     for x ≥ -1 ∧ x < 0  ⎪          ──          \n",
       "           ⎪   6    2    2   6                         ⎪          6           \n",
       "           ⎪                                           ⎪                      \n",
       "           ⎪     3    2                                ⎪   3                  \n",
       "           ⎪    x    x    x   1                        ⎪  x       2         2 \n",
       " ∧ x < 0   ⎪  - ── + ── + ─ + ─    for x ≥ 0 ∧ x < 1   ⎪- ── + 2⋅x  - 2⋅x + ─ \n",
       "           ⎪    2    2    2   6                        ⎪  2                 3 \n",
       "           ⎪                                           ⎪                      \n",
       "         , ⎨  3      2                               , ⎨ 3                    \n",
       "           ⎪ x    5⋅x    7⋅x   5                       ⎪x       2          22 \n",
       " ∧ x < 1   ⎪ ── - ──── + ─── - ─   for x ≥ 1 ∧ x < 2   ⎪── - 4⋅x  + 10⋅x - ── \n",
       "           ⎪ 2     2      2    6                       ⎪2                  3  \n",
       "           ⎪                                           ⎪                      \n",
       "           ⎪   3      2                                ⎪   3                  \n",
       "           ⎪  x    3⋅x    9⋅x   9                      ⎪  x       2         32\n",
       " ∧ x ≤ 2   ⎪- ── + ──── - ─── + ─  for x ≥ 2 ∧ x ≤ 3   ⎪- ── + 2⋅x  - 8⋅x + ──\n",
       "           ⎪  6     2      2    2                      ⎪  6                 3 \n",
       "           ⎪                                           ⎪                      \n",
       "wise       ⎩          0                otherwise       ⎩          0           \n",
       "\n",
       "                     ⎧     3    2                                  ⎧    3     \n",
       "                     ⎪    x    x    x   1                          ⎪   x     2\n",
       "  for x ≥ 0 ∧ x < 1  ⎪    ── - ── + ─ - ─       for x ≥ 1 ∧ x < 2  ⎪   ── - x \n",
       "                     ⎪    6    2    2   6                          ⎪   6      \n",
       "                     ⎪                                             ⎪          \n",
       "                     ⎪   3      2                                  ⎪   3      \n",
       "                     ⎪  x    7⋅x    15⋅x   31                      ⎪  x       \n",
       "  for x ≥ 1 ∧ x < 2  ⎪- ── + ──── - ──── + ──   for x ≥ 2 ∧ x < 3  ⎪- ── + 5⋅x\n",
       "                     ⎪  2     2      2     6                       ⎪  2       \n",
       "                     ⎪                                             ⎪          \n",
       "                   , ⎨ 3       2                                 , ⎨ 3        \n",
       "                     ⎪x    11⋅x    39⋅x   131                      ⎪x       2 \n",
       "  for x ≥ 2 ∧ x < 3  ⎪── - ───── + ──── - ───   for x ≥ 3 ∧ x < 4  ⎪── - 7⋅x  \n",
       "                     ⎪2      2      2      6                       ⎪2         \n",
       "                     ⎪                                             ⎪          \n",
       "                     ⎪   3      2                                  ⎪   3      \n",
       "                     ⎪  x    5⋅x    25⋅x   125                     ⎪  x       \n",
       "  for x ≥ 3 ∧ x ≤ 4  ⎪- ── + ──── - ──── + ───  for x ≥ 4 ∧ x ≤ 5  ⎪- ── + 3⋅x\n",
       "                     ⎪  6     2      2      6                      ⎪  6       \n",
       "                     ⎪                                             ⎪          \n",
       "      otherwise      ⎩           0                  otherwise      ⎩          \n",
       "\n",
       "                                  ⎧    3      2                               \n",
       "         4                        ⎪   x    3⋅x    9⋅x   9                     \n",
       " + 2⋅x - ─     for x ≥ 2 ∧ x < 3  ⎪   ── - ──── + ─── - ─     for x ≥ 3 ∧ x < \n",
       "         3                        ⎪   6     2      2    2                     \n",
       "                                  ⎪                                           \n",
       "                                  ⎪   3       2                               \n",
       "2          50                     ⎪  x    13⋅x    55⋅x   229                  \n",
       "  - 16⋅x + ──  for x ≥ 3 ∧ x < 4  ⎪- ── + ───── - ──── + ───  for x ≥ 4 ∧ x < \n",
       "           3                      ⎪  2      2      2      6                   \n",
       "                                  ⎪                                           \n",
       "                                , ⎨  3       2                                \n",
       "         142                      ⎪ x    17⋅x    95⋅x   521                   \n",
       "+ 32⋅x - ───   for x ≥ 4 ∧ x < 5  ⎪ ── - ───── + ──── - ───   for x ≥ 5 ∧ x < \n",
       "          3                       ⎪ 2      2      2      6                    \n",
       "                                  ⎪                                           \n",
       "                                  ⎪   3      2                                \n",
       "2                                 ⎪  x    7⋅x    49⋅x   343                   \n",
       "  - 18⋅x + 36  for x ≥ 5 ∧ x ≤ 6  ⎪- ── + ──── - ──── + ───   for x ≥ 6 ∧ x ≤ \n",
       "                                  ⎪  6     2      2      6                    \n",
       "                                  ⎪                                           \n",
       " 0                 otherwise      ⎩            0                  otherwise   \n",
       "\n",
       "   ⎧   3                                       ⎤\n",
       "   ⎪  x       2         32                     ⎥\n",
       "4  ⎪  ── - 2⋅x  + 8⋅x - ──    for x ≥ 4 ∧ x < 5⎥\n",
       "   ⎪  6                 3                      ⎥\n",
       "   ⎪                                           ⎥\n",
       "   ⎪   3                                       ⎥\n",
       "   ⎪  x       2          218                   ⎥\n",
       "5  ⎪- ── + 8⋅x  - 42⋅x + ───  for x ≥ 5 ∧ x < 6⎥\n",
       "   ⎪  2                   3                    ⎥\n",
       "   ⎪                                           ⎥\n",
       " , ⎨ 3                                         ⎥\n",
       "   ⎪x        2          430                    ⎥\n",
       "6  ⎪── - 10⋅x  + 66⋅x - ───   for x ≥ 6 ∧ x < 7⎥\n",
       "   ⎪2                    3                     ⎥\n",
       "   ⎪                                           ⎥\n",
       "   ⎪   3                                       ⎥\n",
       "   ⎪  x       2          256                   ⎥\n",
       "7  ⎪- ── + 4⋅x  - 32⋅x + ───  for x ≥ 7 ∧ x ≤ 8⎥\n",
       "   ⎪  6                   3                    ⎥\n",
       "   ⎪                                           ⎥\n",
       "   ⎩           0                  otherwise    ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Need knot values outside the target interval to generate the right set of basis functions.\n",
    "knots = [0,1,2,3,4,5]\n",
    "all_knots = [-3,-2,-1,0,1,2,3,4,5,6,7,8]\n",
    "\n",
    "# Third-order bspline\n",
    "sym_basis = bspline_basis_set(3, all_knots, xs)\n",
    "print(\"Number of basis functions = \",len(sym_basis))\n",
    "sym_basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To approximate a function we need coefficients for each basis function \n",
    "$$\n",
    "f(x) = \\sum_0^{M+2} c_i B_i(x)\n",
    "$$\n",
    "The values of the function at the knots provides $M$ constraints.  We still need 2 more to fully specify the coefficients.   This is where the boundary conditions come into play.\n",
    "\n",
    "$$\n",
    "\\sum_0^{M+2} c_i B_i(x_j) = g(x_j)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAT0AAAAUBAMAAAANYffkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3q6uDrpgLrP8QsMNja5bGOHkL9ljwVrZ/xOYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 1, \\quad -2, \\quad 1, \\quad 0, \\quad 0, \\quad 0, \\quad 0, \\quad 0\\right ]$$"
      ],
      "text/plain": [
       "[1, -2, 1, 0, 0, 0, 0, 0]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAT0AAAAUBAMAAAANYffkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHmld8xQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ 0, \\quad 0, \\quad 0, \\quad 0, \\quad 0, \\quad 1, \\quad -2, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[0, 0, 0, 0, 0, 1, -2, 1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASYAAADICAMAAAByI7yuAAAATlBMVEX///8AAAAAAAAAAAAAAAAA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\noVP8H50KGULXv2+dAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & -2 & 1 & 0 & 0 & 0 & 0 & 0\\\\\\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6} & 0 & 0 & 0 & 0 & 0\\\\0 & \\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6} & 0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6} & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6} & 0 & 0\\\\0 & 0 & 0 & 0 & \\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6} & 0\\\\0 & 0 & 0 & 0 & 0 & \\frac{1}{6} & \\frac{2}{3} & \\frac{1}{6}\\\\0 & 0 & 0 & 0 & 0 & 1 & -2 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 1   -2    1    0    0    0    0    0 ⎤\n",
       "⎢                                      ⎥\n",
       "⎢1/6  2/3  1/6   0    0    0    0    0 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ 0   1/6  2/3  1/6   0    0    0    0 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ 0    0   1/6  2/3  1/6   0    0    0 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ 0    0    0   1/6  2/3  1/6   0    0 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ 0    0    0    0   1/6  2/3  1/6   0 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ 0    0    0    0    0   1/6  2/3  1/6⎥\n",
       "⎢                                      ⎥\n",
       "⎣ 0    0    0    0    0    1   -2    1 ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fill out the matrix where each entry is a basis function evaluated at the knots\n",
    "# The columns are each basis function evaluated at all the knots.\n",
    "mat = Matrix.eye(len(knots)+2)\n",
    "for i,k in enumerate(knots):\n",
    "    for j,basis in enumerate(sym_basis):\n",
    "        bv = basis.subs(xs,k)\n",
    "        mat[i+1,j] = bv\n",
    "\n",
    "# Natural boundary conditions - set 2nd derivative at end of range to zero\n",
    "dd_spline = [diff(bs, xs, 2) for bs in sym_basis]\n",
    "\n",
    "row0 = [dds.subs(xs, 0) for dds in dd_spline]\n",
    "rowN = [dds.subs(xs, knots[-1]) for dds in dd_spline]\n",
    "display(row0)\n",
    "display(rowN)\n",
    "for i in range(len(row0)):\n",
    "    mat[0,i] = row0[i]\n",
    "    mat[-1,i] = rowN[i]\n",
    "mat\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAAUBAMAAABBrq01AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ 0, \\quad 1, \\quad 4, \\quad 9, \\quad 16, \\quad 25\\right ]$$"
      ],
      "text/plain": [
       "[0, 1, 4, 9, 16, 25]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let us assume a simple quadratic function for interpolation\n",
    "func_to_approx = [k*k for k in knots]\n",
    "func_to_approx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve for coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAccAAAAyBAMAAADb+g2fAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ - \\frac{11}{19}, \\quad 0, \\quad \\frac{11}{19}, \\quad \\frac{70}{19}, \\quad \\frac{165}{19}, \\quad \\frac{296}{19}, \\quad 25, \\quad \\frac{654}{19}\\right ]$$"
      ],
      "text/plain": [
       "⎡-11      11  70  165  296      654⎤\n",
       "⎢────, 0, ──, ──, ───, ───, 25, ───⎥\n",
       "⎣ 19      19  19   19   19       19⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs_vals = [0] + func_to_approx + [0]  # Zeros are the value of the second derivative\n",
    "coeffs = mat.LUsolve(Matrix(len(lhs_vals), 1, lhs_vals))\n",
    "coeffs.T.tolist()[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluate the spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def spline_eval(basis, coeffs, x):\n",
    "    val = 0.0\n",
    "    for c,bs in zip(coeffs, basis):\n",
    "        val += c*bs.subs(xs,x)\n",
    "    return val"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(0, 0, 0)\n",
      "(1, 1, 1)\n",
      "(2, 4, 4)\n",
      "(3, 9, 9)\n",
      "(4, 16, 16)\n",
      "(5, 25, 25)\n"
     ]
    }
   ],
   "source": [
    "# check that it reproduces the knots\n",
    "for k,v in zip(knots,func_to_approx):\n",
    "    print(k,spline_eval(sym_basis, coeffs, k),v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fedaf183c10>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Lq5l2fRpn9U3yOpL4kQpdJES8t24Xt89dSXxMJK9OPpX+ydrIOdyo0EWCnHOO5z79mj8u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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fedaf26ba10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Now check elsewhere\n",
    "xvals = []\n",
    "yvals = []\n",
    "for i in range(50):\n",
    "    x = .1*i\n",
    "    val = spline_eval(sym_basis, coeffs, x)\n",
    "    xvals.append(x)\n",
    "    yvals.append(val)\n",
    "plt.plot(xvals, yvals)\n",
    "#plt.plot(xvals, yvals, knots, func_to_approx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matching Einspline evaluation\n",
    "The einspline code collects the values of each power of x in each interval.  The current form of the basis functions is a piecewise interval on the inside, and multiplication by the coefficients on the outside.  We would like to transpose this representation so that the intervals are on the outside, and the constributions from each coefficient are on the inside.  This will require some examination of the Sympy representation for intervals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def to_interval(ival):\n",
    "    \"\"\"Convert relational expression to an Interval\"\"\"\n",
    "    min_val = None\n",
    "    lower_open = False\n",
    "    max_val = None\n",
    "    upper_open = True\n",
    "    if isinstance(ival, And):\n",
    "        for rel in ival.args:\n",
    "            #print('rel ',rel, type(rel), rel.args[1])\n",
    "            if isinstance(rel, StrictGreaterThan):\n",
    "                min_val = rel.args[1]\n",
    "                #lower_open = True\n",
    "            elif isinstance(rel, GreaterThan):\n",
    "                min_val = rel.args[1]\n",
    "                #lower_open = False\n",
    "            elif isinstance(rel, StrictLessThan):\n",
    "                max_val = rel.args[1]\n",
    "                #upper_open = True\n",
    "            elif isinstance(rel, LessThan):\n",
    "                max_val = rel.args[1]\n",
    "                #upper_open = False\n",
    "            else:\n",
    "                print('unhandled ',rel)\n",
    "\n",
    "    if min_val == None or max_val == None:\n",
    "        print('error',ival)\n",
    "    return Interval(min_val, max_val, lower_open, upper_open)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(0, 1))\n"
     ]
    },
    {
     "data": {
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      "text/latex": [
       "$$\\left [ \\left ( 0, \\quad - \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - \\frac{x}{2} + \\frac{1}{6}\\right ), \\quad \\left ( 1, \\quad \\frac{x^{3}}{2} - x^{2} + \\frac{2}{3}\\right ), \\quad \\left ( 2, \\quad - \\frac{x^{3}}{2} + \\frac{x^{2}}{2} + \\frac{x}{2} + \\frac{1}{6}\\right ), \\quad \\left ( 3, \\quad \\frac{x^{3}}{6}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3    2        ⎞  ⎛    3         ⎞  ⎛      3    2        ⎞  ⎛    3⎞⎤\n",
       "⎢⎜     x    x    x   1⎟  ⎜   x     2   2⎟  ⎜     x    x    x   1⎟  ⎜   x ⎟⎥\n",
       "⎢⎜0, - ── + ── - ─ + ─⎟, ⎜1, ── - x  + ─⎟, ⎜2, - ── + ── + ─ + ─⎟, ⎜3, ──⎟⎥\n",
       "⎣⎝     6    2    2   6⎠  ⎝   2         3⎠  ⎝     2    2    2   6⎠  ⎝   6 ⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(1, 2))\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left ( 1, \\quad - \\frac{x^{3}}{6} + x^{2} - 2 x + \\frac{4}{3}\\right ), \\quad \\left ( 2, \\quad \\frac{x^{3}}{2} - \\frac{5 x^{2}}{2} + \\frac{7 x}{2} - \\frac{5}{6}\\right ), \\quad \\left ( 3, \\quad - \\frac{x^{3}}{2} + 2 x^{2} - 2 x + \\frac{2}{3}\\right ), \\quad \\left ( 4, \\quad \\frac{x^{3}}{6} - \\frac{x^{2}}{2} + \\frac{x}{2} - \\frac{1}{6}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3               ⎞  ⎛    3      2          ⎞  ⎛      3                 \n",
       "⎢⎜     x     2         4⎟  ⎜   x    5⋅x    7⋅x   5⎟  ⎜     x       2         2\n",
       "⎢⎜1, - ── + x  - 2⋅x + ─⎟, ⎜2, ── - ──── + ─── - ─⎟, ⎜3, - ── + 2⋅x  - 2⋅x + ─\n",
       "⎣⎝     6               3⎠  ⎝   2     2      2    6⎠  ⎝     2                 3\n",
       "\n",
       "⎞  ⎛    3    2        ⎞⎤\n",
       "⎟  ⎜   x    x    x   1⎟⎥\n",
       "⎟, ⎜4, ── - ── + ─ - ─⎟⎥\n",
       "⎠  ⎝   6    2    2   6⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(3, 4))\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left ( 3, \\quad - \\frac{x^{3}}{6} + 2 x^{2} - 8 x + \\frac{32}{3}\\right ), \\quad \\left ( 4, \\quad \\frac{x^{3}}{2} - \\frac{11 x^{2}}{2} + \\frac{39 x}{2} - \\frac{131}{6}\\right ), \\quad \\left ( 5, \\quad - \\frac{x^{3}}{2} + 5 x^{2} - 16 x + \\frac{50}{3}\\right ), \\quad \\left ( 6, \\quad \\frac{x^{3}}{6} - \\frac{3 x^{2}}{2} + \\frac{9 x}{2} - \\frac{9}{2}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3                  ⎞  ⎛    3       2             ⎞  ⎛      3          \n",
       "⎢⎜     x       2         32⎟  ⎜   x    11⋅x    39⋅x   131⎟  ⎜     x       2   \n",
       "⎢⎜3, - ── + 2⋅x  - 8⋅x + ──⎟, ⎜4, ── - ───── + ──── - ───⎟, ⎜5, - ── + 5⋅x  - \n",
       "⎣⎝     6                 3 ⎠  ⎝   2      2      2      6 ⎠  ⎝     2           \n",
       "\n",
       "         ⎞  ⎛    3      2          ⎞⎤\n",
       "       50⎟  ⎜   x    3⋅x    9⋅x   9⎟⎥\n",
       "16⋅x + ──⎟, ⎜6, ── - ──── + ─── - ─⎟⎥\n",
       "       3 ⎠  ⎝   6     2      2    2⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(2, 3))\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left ( 2, \\quad - \\frac{x^{3}}{6} + \\frac{3 x^{2}}{2} - \\frac{9 x}{2} + \\frac{9}{2}\\right ), \\quad \\left ( 3, \\quad \\frac{x^{3}}{2} - 4 x^{2} + 10 x - \\frac{22}{3}\\right ), \\quad \\left ( 4, \\quad - \\frac{x^{3}}{2} + \\frac{7 x^{2}}{2} - \\frac{15 x}{2} + \\frac{31}{6}\\right ), \\quad \\left ( 5, \\quad \\frac{x^{3}}{6} - x^{2} + 2 x - \\frac{4}{3}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3      2          ⎞  ⎛    3                   ⎞  ⎛      3      2      \n",
       "⎢⎜     x    3⋅x    9⋅x   9⎟  ⎜   x       2          22⎟  ⎜     x    7⋅x    15⋅\n",
       "⎢⎜2, - ── + ──── - ─── + ─⎟, ⎜3, ── - 4⋅x  + 10⋅x - ──⎟, ⎜4, - ── + ──── - ───\n",
       "⎣⎝     6     2      2    2⎠  ⎝   2                  3 ⎠  ⎝     2     2      2 \n",
       "\n",
       "      ⎞  ⎛    3               ⎞⎤\n",
       "x   31⎟  ⎜   x     2         4⎟⎥\n",
       "─ + ──⎟, ⎜5, ── - x  + 2⋅x - ─⎟⎥\n",
       "    6 ⎠  ⎝   6               3⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(4, 5))\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left ( 4, \\quad - \\frac{x^{3}}{6} + \\frac{5 x^{2}}{2} - \\frac{25 x}{2} + \\frac{125}{6}\\right ), \\quad \\left ( 5, \\quad \\frac{x^{3}}{2} - 7 x^{2} + 32 x - \\frac{142}{3}\\right ), \\quad \\left ( 6, \\quad - \\frac{x^{3}}{2} + \\frac{13 x^{2}}{2} - \\frac{55 x}{2} + \\frac{229}{6}\\right ), \\quad \\left ( 7, \\quad \\frac{x^{3}}{6} - 2 x^{2} + 8 x - \\frac{32}{3}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3      2             ⎞  ⎛    3                    ⎞  ⎛      3       2 \n",
       "⎢⎜     x    5⋅x    25⋅x   125⎟  ⎜   x       2          142⎟  ⎜     x    13⋅x  \n",
       "⎢⎜4, - ── + ──── - ──── + ───⎟, ⎜5, ── - 7⋅x  + 32⋅x - ───⎟, ⎜6, - ── + ───── \n",
       "⎣⎝     6     2      2      6 ⎠  ⎝   2                   3 ⎠  ⎝     2      2   \n",
       "\n",
       "            ⎞  ⎛    3                  ⎞⎤\n",
       "  55⋅x   229⎟  ⎜   x       2         32⎟⎥\n",
       "- ──── + ───⎟, ⎜7, ── - 2⋅x  + 8⋅x - ──⎟⎥\n",
       "   2      6 ⎠  ⎝   6                 3 ⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Interval = ', Interval.Ropen(5, 6))\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left ( 5, \\quad - \\frac{x^{3}}{6} + 3 x^{2} - 18 x + 36\\right ), \\quad \\left ( 6, \\quad \\frac{x^{3}}{2} - \\frac{17 x^{2}}{2} + \\frac{95 x}{2} - \\frac{521}{6}\\right ), \\quad \\left ( 7, \\quad - \\frac{x^{3}}{2} + 8 x^{2} - 42 x + \\frac{218}{3}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛      3                   ⎞  ⎛    3       2             ⎞  ⎛      3         \n",
       "⎢⎜     x       2            ⎟  ⎜   x    17⋅x    95⋅x   521⎟  ⎜     x       2  \n",
       "⎢⎜5, - ── + 3⋅x  - 18⋅x + 36⎟, ⎜6, ── - ───── + ──── - ───⎟, ⎜7, - ── + 8⋅x  -\n",
       "⎣⎝     6                    ⎠  ⎝   2      2      2      6 ⎠  ⎝     2          \n",
       "\n",
       "           ⎞⎤\n",
       "        218⎟⎥\n",
       " 42⋅x + ───⎟⎥\n",
       "         3 ⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Transpose the interval and coefficients\n",
    "#  Note that interval [0,1) has the polynomial coefficients found in the einspline code\n",
    "#  The other intervals could be shifted, and they would also have the same polynomials\n",
    "def transpose_interval_and_coefficients(sym_basis):\n",
    "    cond_map = defaultdict(list)\n",
    "\n",
    "    i1 = Interval(0,5, False, False) # interval for evaluation\n",
    "    for idx, s0 in enumerate(sym_basis):\n",
    "        for expr, cond in s0.args:\n",
    "            if cond != True:\n",
    "                i2 = to_interval(cond)\n",
    "                if not i1.is_disjoint(i2):\n",
    "                    cond_map[i2].append( (idx, expr) )\n",
    "    return cond_map\n",
    "\n",
    "cond_map = transpose_interval_and_coefficients(sym_basis)\n",
    "for cond, expr in cond_map.items():\n",
    "    #print(cond, [e.subs(x, x-cond.args[0]) for e in expr])\n",
    "    print(\"Interval = \",cond)\n",
    "    # Shift interval to a common start - see that the polynomial coefficients are all the same\n",
    "    #e2 = [expand(e[1].subs(xs, xs+cond.args[0])) for e in expr]\n",
    "    e2 = [(idx,expand(e)) for idx, e in expr]\n",
    "    display(e2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qbEpTCbYKnOLy5Wi3gVV6ywhpDfn+33w26mo/fKx/XT6c2wc//gF7/Cor7AD2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\n6vnr7WOLjJuxJOUtWG4ijO/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43RoD7tVirgqRA21AKuTBmwXvljUOzualvyO+FXYM85G5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\nkXH1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\begin{cases} \\frac{x^{3} c_{3}}{6} + \\left(\\frac{x^{3}}{2} - x^{2} + \\frac{2}{3}\\right) c_{1} + \\left(- \\frac{x^{3}}{2} + \\frac{x^{2}}{2} + \\frac{x}{2} + \\frac{1}{6}\\right) c_{2} + \\left(- \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - \\frac{x}{2} + \\frac{1}{6}\\right) c_{0} & \\text{for}\\: 0 \\leq x \\wedge x < 1 \\\\\\left(- \\frac{x^{3}}{2} + 2 x^{2} - 2 x + \\frac{2}{3}\\right) c_{3} + \\left(- \\frac{x^{3}}{6} + x^{2} - 2 x + \\frac{4}{3}\\right) c_{1} + \\left(\\frac{x^{3}}{6} - \\frac{x^{2}}{2} + \\frac{x}{2} - \\frac{1}{6}\\right) c_{4} + \\left(\\frac{x^{3}}{2} - \\frac{5 x^{2}}{2} + \\frac{7 x}{2} - \\frac{5}{6}\\right) c_{2} & \\text{for}\\: 1 \\leq x \\wedge x < 2 \\\\\\left(- \\frac{x^{3}}{2} + 5 x^{2} - 16 x + \\frac{50}{3}\\right) c_{5} + \\left(- \\frac{x^{3}}{6} + 2 x^{2} - 8 x + \\frac{32}{3}\\right) c_{3} + \\left(\\frac{x^{3}}{6} - \\frac{3 x^{2}}{2} + \\frac{9 x}{2} - \\frac{9}{2}\\right) c_{6} + \\left(\\frac{x^{3}}{2} - \\frac{11 x^{2}}{2} + \\frac{39 x}{2} - \\frac{131}{6}\\right) c_{4} & \\text{for}\\: 3 \\leq x \\wedge x < 4 \\\\\\left(- \\frac{x^{3}}{2} + \\frac{7 x^{2}}{2} - \\frac{15 x}{2} + \\frac{31}{6}\\right) c_{4} + \\left(- \\frac{x^{3}}{6} + \\frac{3 x^{2}}{2} - \\frac{9 x}{2} + \\frac{9}{2}\\right) c_{2} + \\left(\\frac{x^{3}}{6} - x^{2} + 2 x - \\frac{4}{3}\\right) c_{5} + \\left(\\frac{x^{3}}{2} - 4 x^{2} + 10 x - \\frac{22}{3}\\right) c_{3} & \\text{for}\\: 2 \\leq x \\wedge x < 3 \\\\\\left(- \\frac{x^{3}}{2} + \\frac{13 x^{2}}{2} - \\frac{55 x}{2} + \\frac{229}{6}\\right) c_{6} + \\left(- \\frac{x^{3}}{6} + \\frac{5 x^{2}}{2} - \\frac{25 x}{2} + \\frac{125}{6}\\right) c_{4} + \\left(\\frac{x^{3}}{6} - 2 x^{2} + 8 x - \\frac{32}{3}\\right) c_{7} + \\left(\\frac{x^{3}}{2} - 7 x^{2} + 32 x - \\frac{142}{3}\\right) c_{5} & \\text{for}\\: 4 \\leq x \\wedge x < 5 \\\\\\left(- \\frac{x^{3}}{2} + 8 x^{2} - 42 x + \\frac{218}{3}\\right) c_{7} + \\left(- \\frac{x^{3}}{6} + 3 x^{2} - 18 x + 36\\right) c_{5} + \\left(\\frac{x^{3}}{2} - \\frac{17 x^{2}}{2} + \\frac{95 x}{2} - \\frac{521}{6}\\right) c_{6} & \\text{for}\\: 5 \\leq x \\wedge x < 6 \\end{cases}$$"
      ],
      "text/plain": [
       "⎧                        3        ⎛ 3         ⎞        ⎛   3    2        ⎞    \n",
       "⎪                       x ⋅c[3]   ⎜x     2   2⎟        ⎜  x    x    x   1⎟    \n",
       "⎪                       ─────── + ⎜── - x  + ─⎟⋅c[1] + ⎜- ── + ── + ─ + ─⎟⋅c[2\n",
       "⎪                          6      ⎝2         3⎠        ⎝  2    2    2   6⎠    \n",
       "⎪                                                                             \n",
       "⎪        ⎛   3                 ⎞        ⎛   3               ⎞        ⎛ 3    2 \n",
       "⎪        ⎜  x       2         2⎟        ⎜  x     2         4⎟        ⎜x    x  \n",
       "⎪        ⎜- ── + 2⋅x  - 2⋅x + ─⎟⋅c[3] + ⎜- ── + x  - 2⋅x + ─⎟⋅c[1] + ⎜── - ── \n",
       "⎪        ⎝  2                 3⎠        ⎝  6               3⎠        ⎝6    2  \n",
       "⎪                                                                             \n",
       "⎪  ⎛   3                   ⎞        ⎛   3                  ⎞        ⎛ 3      2\n",
       "⎪  ⎜  x       2          50⎟        ⎜  x       2         32⎟        ⎜x    3⋅x \n",
       "⎪  ⎜- ── + 5⋅x  - 16⋅x + ──⎟⋅c[5] + ⎜- ── + 2⋅x  - 8⋅x + ──⎟⋅c[3] + ⎜── - ────\n",
       "⎪  ⎝  2                  3 ⎠        ⎝  6                 3 ⎠        ⎝6     2  \n",
       "⎨                                                                             \n",
       "⎪    ⎛   3      2            ⎞        ⎛   3      2          ⎞        ⎛ 3      \n",
       "⎪    ⎜  x    7⋅x    15⋅x   31⎟        ⎜  x    3⋅x    9⋅x   9⎟        ⎜x     2 \n",
       "⎪    ⎜- ── + ──── - ──── + ──⎟⋅c[4] + ⎜- ── + ──── - ─── + ─⎟⋅c[2] + ⎜── - x  \n",
       "⎪    ⎝  2     2      2     6 ⎠        ⎝  6     2      2    2⎠        ⎝6       \n",
       "⎪                                                                             \n",
       "⎪⎛   3       2             ⎞        ⎛   3      2             ⎞        ⎛ 3     \n",
       "⎪⎜  x    13⋅x    55⋅x   229⎟        ⎜  x    5⋅x    25⋅x   125⎟        ⎜x      \n",
       "⎪⎜- ── + ───── - ──── + ───⎟⋅c[6] + ⎜- ── + ──── - ──── + ───⎟⋅c[4] + ⎜── - 2⋅\n",
       "⎪⎝  2      2      2      6 ⎠        ⎝  6     2      2      6 ⎠        ⎝6      \n",
       "⎪                                                                             \n",
       "⎪               ⎛   3                    ⎞        ⎛   3                   ⎞   \n",
       "⎪               ⎜  x       2          218⎟        ⎜  x       2            ⎟   \n",
       "⎪               ⎜- ── + 8⋅x  - 42⋅x + ───⎟⋅c[7] + ⎜- ── + 3⋅x  - 18⋅x + 36⎟⋅c[\n",
       "⎩               ⎝  2                   3 ⎠        ⎝  6                    ⎠   \n",
       "\n",
       "    ⎛   3    2        ⎞                                               \n",
       "    ⎜  x    x    x   1⎟                                               \n",
       "] + ⎜- ── + ── - ─ + ─⎟⋅c[0]                         for 0 ≤ x ∧ x < 1\n",
       "    ⎝  6    2    2   6⎠                                               \n",
       "                                                                      \n",
       "       ⎞        ⎛ 3      2          ⎞                                 \n",
       "  x   1⎟        ⎜x    5⋅x    7⋅x   5⎟                                 \n",
       "+ ─ - ─⎟⋅c[4] + ⎜── - ──── + ─── - ─⎟⋅c[2]           for 1 ≤ x ∧ x < 2\n",
       "  2   6⎠        ⎝2     2      2    6⎠                                 \n",
       "                                                                      \n",
       "          ⎞        ⎛ 3       2             ⎞                          \n",
       "   9⋅x   9⎟        ⎜x    11⋅x    39⋅x   131⎟                          \n",
       " + ─── - ─⎟⋅c[6] + ⎜── - ───── + ──── - ───⎟⋅c[4]    for 3 ≤ x ∧ x < 4\n",
       "    2    2⎠        ⎝2      2      2      6 ⎠                          \n",
       "                                                                      \n",
       "         ⎞        ⎛ 3                   ⎞                             \n",
       "        4⎟        ⎜x       2          22⎟                             \n",
       "+ 2⋅x - ─⎟⋅c[5] + ⎜── - 4⋅x  + 10⋅x - ──⎟⋅c[3]       for 2 ≤ x ∧ x < 3\n",
       "        3⎠        ⎝2                  3 ⎠                             \n",
       "                                                                      \n",
       "             ⎞        ⎛ 3                    ⎞                        \n",
       " 2         32⎟        ⎜x       2          142⎟                        \n",
       "x  + 8⋅x - ──⎟⋅c[7] + ⎜── - 7⋅x  + 32⋅x - ───⎟⋅c[5]  for 4 ≤ x ∧ x < 5\n",
       "           3 ⎠        ⎝2                   3 ⎠                        \n",
       "                                                                      \n",
       "     ⎛ 3       2             ⎞                                        \n",
       "     ⎜x    17⋅x    95⋅x   521⎟                                        \n",
       "5] + ⎜── - ───── + ──── - ───⎟⋅c[6]                  for 5 ≤ x ∧ x < 6\n",
       "     ⎝2      2      2      6 ⎠                                        "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create piecewise expression from the transposed intervals\n",
    "def recreate_piecewise(basis_map, c):\n",
    "    args = []\n",
    "    for cond, exprs in basis_map.items():\n",
    "        e = 0\n",
    "        for idx, b in exprs:\n",
    "            e += c[idx] * b\n",
    "        args.append( (e, cond.as_relational(xs)))\n",
    "    return Piecewise(*args)\n",
    "\n",
    "c = IndexedBase('c')\n",
    "spline = recreate_piecewise(cond_map, c)\n",
    "spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def spline_eval2(spline, coeffs, x):\n",
    "    \"\"\"Evaluate spline using transposed expression\"\"\"\n",
    "    val = 0.0\n",
    "    c = IndexedBase('c')\n",
    "    to_sub = {}\n",
    "    for i,cf in enumerate(coeffs):\n",
    "        to_sub[c[i]] = cf\n",
    "    to_sub[xs] = x\n",
    "    return spline.subs(to_sub)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(0, 0)\n",
      "(1, 1)\n",
      "(2, 4)\n",
      "(3, 9)\n",
      "(4, 16)\n",
      "(5, 25)\n"
     ]
    }
   ],
   "source": [
    "for k in knots:\n",
    "    val = spline_eval2(spline, coeffs, k)\n",
    "    print(k,val)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Finding coefficients for periodic functions\n",
    "\n",
    "Because of periodicity, the last coefficients will have the same values as the first coefficients. \n",
    "If there are 5 knots, there will be 8 coefficients, but $c[5] = c[0]$, $c[6] = c[1]$, and $c[7] = c[2]$.  There only needs to same number of equations as the number of knots.\n",
    "\n",
    "We need to generate the same matrix equation for values.  This time we will use the full spline expression and manipulate the coefficient array to pick out the values needed.  This is a slightly different form than the non-periodic case, but will produce the same effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix([[1/6, 2/3, 1/6, 0, 0, 0]])\n",
      "Matrix([[0, 1/6, 2/3, 1/6, 0, 0]])\n",
      "Matrix([[0, 0, 1/6, 2/3, 1/6, 0]])\n",
      "Matrix([[0, 0, 0, 1/6, 2/3, 1/6]])\n",
      "Matrix([[1/6, 0, 0, 0, 1/6, 2/3]])\n",
      "Matrix([[2/3, 1/6, 0, 0, 0, 1/6]])\n"
     ]
    }
   ],
   "source": [
    "m = len(knots)\n",
    "mat = Matrix.eye(m)\n",
    "for i in range(m):\n",
    "    for j in range(m):\n",
    "        subslist = {}\n",
    "        for k in range(m+3):\n",
    "            if k%m == i:\n",
    "                subslist[c[k]] = 1\n",
    "            else:\n",
    "                subslist[c[k]] = 0\n",
    "        subslist[xs] = j # *Delta, but for this example Delta=1.0\n",
    "        e = spline.subs(subslist)\n",
    "        mat[j,i] = e\n",
    "        \n",
    "for i in range(m):\n",
    "    print mat.row(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0.0 0\n",
      "1 1.0 0.866025403784439\n",
      "2 2.0 0.866025403784439\n",
      "3 3.0 0\n",
      "4 4.0 -0.866025403784438\n",
      "5 5.0 -0.866025403784439\n"
     ]
    }
   ],
   "source": [
    "# Create the rhs vector of values to interpolate\n",
    "rhs = []\n",
    "xvals = []\n",
    "for i in range(m):\n",
    "    x = i%m*1.0\n",
    "    val = sin(S.Pi*x/3.0).evalf()\n",
    "    print i%m, x, val\n",
    "    rhs.append(val)\n",
    "    xvals.append(x)\n",
    "b = Matrix(rhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}-1.03923048454133\\\\0\\\\1.03923048454133\\\\1.03923048454133\\\\-9.13554945977272 \\cdot 10^{-16}\\\\-1.03923048454132\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  -1.03923048454133  ⎤\n",
       "⎢                     ⎥\n",
       "⎢          0          ⎥\n",
       "⎢                     ⎥\n",
       "⎢  1.03923048454133   ⎥\n",
       "⎢                     ⎥\n",
       "⎢  1.03923048454133   ⎥\n",
       "⎢                     ⎥\n",
       "⎢-9.13554945977272e-16⎥\n",
       "⎢                     ⎥\n",
       "⎣  -1.03923048454132  ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "raw_coefs = mat.LUsolve(b)\n",
    "raw_coefs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Fill out the coefficient array\n",
    "coefs = np.zeros(m+3)\n",
    "for i in range(m):\n",
    "    coefs[i] = raw_coefs[i]\n",
    "coefs[m:m+3] = raw_coefs[0:3]\n",
    "\n",
    "# Fill out data arrays for plotting\n",
    "xvals.append(xvals[-1]+1.0)\n",
    "rhs.append(rhs[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fedaf04c750>,\n",
       " <matplotlib.lines.Line2D at 0x7fedaf04c950>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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dvDfww5Qfyc7Vs6Pq0v2acpDp6/fz906NqJT+B9S5CrzLZtphLQxKlQKf1g+Q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\npZRSRfw/oOxXYbN6pmsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fedaf15aa10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# values of spline\n",
    "sp_xvals = []\n",
    "sp_yvals = []\n",
    "for i in range(61):\n",
    "    x = i/10.0\n",
    "    v = spline_eval2(spline, coefs, x%6.0)\n",
    "    #print x,v\n",
    "    sp_xvals.append(x)\n",
    "    sp_yvals.append(v)\n",
    "    \n",
    "plt.plot(xvals,rhs,sp_xvals,sp_yvals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Bspline for Jastrow\n",
    "\n",
    "For the radial part of the Jastrow factor, the derivative at $r=0$ is fixed by the cusp condition.   At $r=r_{cut}$, the value and derivatives are zero.\n",
    "\n",
    "Also add the grid spacing, $\\Delta$, to the knots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'knots = '"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAScAAAAUBAMAAAA97ebFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/4ZmDtX/Aks2FK6SGv5JAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 0, \\quad \\Delta, \\quad 2 \\Delta, \\quad 3 \\Delta, \\quad 4 \\Delta, \\quad 5 \\Delta\\right ]$$"
      ],
      "text/plain": [
       "[0, Δ, 2⋅Δ, 3⋅Δ, 4⋅Δ, 5⋅Δ]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('Number of basis functions = ', 8)\n"
     ]
    }
   ],
   "source": [
    "Delta = Symbol('Delta',positive=True)\n",
    "#knots = [0,1*Delta,2*Delta,3*Delta,4*Delta,5*Delta]\n",
    "nknots = 6\n",
    "knots = [i*Delta for i in range(nknots)]\n",
    "display('knots = ',knots)\n",
    "#all_knots = [-3*Delta,-2*Delta,-1*Delta,0,1*Delta,2*Delta,3*Delta,4*Delta,5*Delta,6*Delta,7*Delta,8*Delta]\n",
    "all_knots = [i*Delta for i in range(-3,nknots+3)]\n",
    "#display('all knots',all_knots)\n",
    "rcut = (nknots-1)*Delta\n",
    "\n",
    "# Third-order bspline\n",
    "jastrow_sym_basis = bspline_basis_set(3, all_knots, xs)\n",
    "print(\"Number of basis functions = \",len(jastrow_sym_basis))\n",
    "#jastrow_sym_basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WJVhV/wUV3tDybg0CwGrGiXGhsVlfiACpjVXSfXfegMTzJjvQYiiUvOAmxjqd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\nZH99oCLwP0sxj7APi2OuAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\begin{cases} \\left(- \\frac{521}{6} + \\frac{95 x}{2 \\Delta} - \\frac{17 x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{6} + \\left(36 - \\frac{18 x}{\\Delta} + \\frac{3 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{5} + \\left(\\frac{218}{3} - \\frac{42 x}{\\Delta} + \\frac{8 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{7} & \\text{for}\\: 5 \\Delta \\leq x \\wedge x < 6 \\Delta \\\\\\left(- \\frac{131}{6} + \\frac{39 x}{2 \\Delta} - \\frac{11 x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{4} + \\left(- \\frac{9}{2} + \\frac{9 x}{2 \\Delta} - \\frac{3 x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{6} + \\left(\\frac{32}{3} - \\frac{8 x}{\\Delta} + \\frac{2 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{3} + \\left(\\frac{50}{3} - \\frac{16 x}{\\Delta} + \\frac{5 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{5} & \\text{for}\\: 3 \\Delta \\leq x \\wedge x < 4 \\Delta \\\\\\left(- \\frac{430}{3} + \\frac{66 x}{\\Delta} - \\frac{10 x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{7} + \\left(\\frac{343}{6} - \\frac{49 x}{2 \\Delta} + \\frac{7 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{6} & \\text{for}\\: 6 \\Delta \\leq x \\wedge x < 7 \\Delta \\\\\\left(\\frac{2}{3} - \\frac{x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{1} + \\left(\\frac{1}{6} - \\frac{x}{2 \\Delta} + \\frac{x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{0} + \\left(\\frac{1}{6} + \\frac{x}{2 \\Delta} + \\frac{x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{2} + \\frac{x^{3} c_{3}}{6 \\Delta^{3}} & \\text{for}\\: 0 \\leq x \\wedge x < \\Delta \\\\\\left(- \\frac{22}{3} + \\frac{10 x}{\\Delta} - \\frac{4 x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{3} + \\left(- \\frac{4}{3} + \\frac{2 x}{\\Delta} - \\frac{x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{5} + \\left(\\frac{9}{2} - \\frac{9 x}{2 \\Delta} + \\frac{3 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{2} + \\left(\\frac{31}{6} - \\frac{15 x}{2 \\Delta} + \\frac{7 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{4} & \\text{for}\\: 2 \\Delta \\leq x \\wedge x < 3 \\Delta \\\\\\left(- \\frac{5}{6} - \\frac{7 x}{2 \\Delta} - \\frac{5 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{0} + \\left(\\frac{4}{3} + \\frac{2 x}{\\Delta} + \\frac{x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{1} & \\text{for}\\: - 2 \\Delta \\leq x \\wedge x < - \\Delta \\\\\\left(\\frac{2}{3} - \\frac{x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{1} + \\left(\\frac{1}{6} - \\frac{x}{2 \\Delta} + \\frac{x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{0} + \\left(\\frac{1}{6} + \\frac{x}{2 \\Delta} + \\frac{x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{2} & \\text{for}\\: - \\Delta \\leq x \\wedge x < 0 \\\\\\left(\\frac{9}{2} + \\frac{9 x}{2 \\Delta} + \\frac{3 x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{0} & \\text{for}\\: - 3 \\Delta \\leq x \\wedge x < - 2 \\Delta \\\\\\left(- \\frac{142}{3} + \\frac{32 x}{\\Delta} - \\frac{7 x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{5} + \\left(- \\frac{32}{3} + \\frac{8 x}{\\Delta} - \\frac{2 x^{2}}{\\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{7} + \\left(\\frac{125}{6} - \\frac{25 x}{2 \\Delta} + \\frac{5 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{4} + \\left(\\frac{229}{6} - \\frac{55 x}{2 \\Delta} + \\frac{13 x^{2}}{2 \\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{6} & \\text{for}\\: 4 \\Delta \\leq x \\wedge x < 5 \\Delta \\\\\\left(\\frac{256}{3} - \\frac{32 x}{\\Delta} + \\frac{4 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{7} & \\text{for}\\: 7 \\Delta \\leq x \\wedge x < 8 \\Delta \\\\\\left(- \\frac{5}{6} + \\frac{7 x}{2 \\Delta} - \\frac{5 x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{2} + \\left(- \\frac{1}{6} + \\frac{x}{2 \\Delta} - \\frac{x^{2}}{2 \\Delta^{2}} + \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{4} + \\left(\\frac{2}{3} - \\frac{2 x}{\\Delta} + \\frac{2 x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{2 \\Delta^{3}}\\right) c_{3} + \\left(\\frac{4}{3} - \\frac{2 x}{\\Delta} + \\frac{x^{2}}{\\Delta^{2}} - \\frac{x^{3}}{6 \\Delta^{3}}\\right) c_{1} & \\text{for}\\: \\Delta \\leq x \\wedge x < 2 \\Delta \\end{cases}$$"
      ],
      "text/plain": [
       "⎧                 ⎛                   2     3 ⎞        ⎛               2     3\n",
       "⎪                 ⎜  521   95⋅x   17⋅x     x  ⎟        ⎜     18⋅x   3⋅x     x \n",
       "⎪                 ⎜- ─── + ──── - ───── + ────⎟⋅c[6] + ⎜36 - ──── + ──── - ───\n",
       "⎪                 ⎜   6    2⋅Δ        2      3⎟        ⎜      Δ       2       \n",
       "⎪                 ⎝                2⋅Δ    2⋅Δ ⎠        ⎝             Δ     6⋅Δ\n",
       "⎪                                                                             \n",
       "⎪  ⎛                   2     3 ⎞        ⎛               2     3 ⎞        ⎛    \n",
       "⎪  ⎜  131   39⋅x   11⋅x     x  ⎟        ⎜  9   9⋅x   3⋅x     x  ⎟        ⎜32  \n",
       "⎪  ⎜- ─── + ──── - ───── + ────⎟⋅c[4] + ⎜- ─ + ─── - ──── + ────⎟⋅c[6] + ⎜── -\n",
       "⎪  ⎜   6    2⋅Δ        2      3⎟        ⎜  2   2⋅Δ      2      3⎟        ⎜3   \n",
       "⎪  ⎝                2⋅Δ    2⋅Δ ⎠        ⎝            2⋅Δ    6⋅Δ ⎠        ⎝    \n",
       "⎪                                                                             \n",
       "⎪                                  ⎛                   2     3 ⎞        ⎛     \n",
       "⎪                                  ⎜  430   66⋅x   10⋅x     x  ⎟        ⎜343  \n",
       "⎪                                  ⎜- ─── + ──── - ───── + ────⎟⋅c[7] + ⎜─── -\n",
       "⎪                                  ⎜   3     Δ        2       3⎟        ⎜ 6   \n",
       "⎪                                  ⎝                 Δ     2⋅Δ ⎠        ⎝     \n",
       "⎪                                                                             \n",
       "⎪                      ⎛     2     3 ⎞        ⎛            2      3 ⎞        ⎛\n",
       "⎪                      ⎜2   x     x  ⎟        ⎜1    x     x      x  ⎟        ⎜\n",
       "⎪                      ⎜─ - ── + ────⎟⋅c[1] + ⎜─ - ─── + ──── - ────⎟⋅c[0] + ⎜\n",
       "⎪                      ⎜3    2      3⎟        ⎜6   2⋅Δ      2      3⎟        ⎜\n",
       "⎪                      ⎝    Δ    2⋅Δ ⎠        ⎝          2⋅Δ    6⋅Δ ⎠        ⎝\n",
       "⎪                                                                             \n",
       "⎪    ⎛                 2     3 ⎞        ⎛             2     3 ⎞        ⎛      \n",
       "⎪    ⎜  22   10⋅x   4⋅x     x  ⎟        ⎜  4   2⋅x   x     x  ⎟        ⎜9   9⋅\n",
       "⎪    ⎜- ── + ──── - ──── + ────⎟⋅c[3] + ⎜- ─ + ─── - ── + ────⎟⋅c[5] + ⎜─ - ──\n",
       "⎪    ⎜  3     Δ       2       3⎟        ⎜  3    Δ     2      3⎟        ⎜2   2⋅\n",
       "⎪    ⎝               Δ     2⋅Δ ⎠        ⎝            Δ    6⋅Δ ⎠        ⎝      \n",
       "⎪                                                                             \n",
       "⎪                                      ⎛               2     3 ⎞        ⎛     \n",
       "⎪                                      ⎜  5   7⋅x   5⋅x     x  ⎟        ⎜4   2\n",
       "⎨                                      ⎜- ─ - ─── - ──── - ────⎟⋅c[0] + ⎜─ + ─\n",
       "⎪                                      ⎜  6   2⋅Δ      2      3⎟        ⎜3    \n",
       "⎪                                      ⎝            2⋅Δ    2⋅Δ ⎠        ⎝     \n",
       "⎪                                                                             \n",
       "⎪                           ⎛     2     3 ⎞        ⎛            2      3 ⎞    \n",
       "⎪                           ⎜2   x     x  ⎟        ⎜1    x     x      x  ⎟    \n",
       "⎪                           ⎜─ - ── - ────⎟⋅c[1] + ⎜─ - ─── + ──── + ────⎟⋅c[0\n",
       "⎪                           ⎜3    2      3⎟        ⎜6   2⋅Δ      2      3⎟    \n",
       "⎪                           ⎝    Δ    2⋅Δ ⎠        ⎝          2⋅Δ    2⋅Δ ⎠    \n",
       "⎪                                                                             \n",
       "⎪                                                      ⎛             2     3 ⎞\n",
       "⎪                                                      ⎜9   9⋅x   3⋅x     x  ⎟\n",
       "⎪                                                      ⎜─ + ─── + ──── + ────⎟\n",
       "⎪                                                      ⎜2   2⋅Δ      2      3⎟\n",
       "⎪                                                      ⎝          2⋅Δ    6⋅Δ ⎠\n",
       "⎪                                                                             \n",
       "⎪⎛                  2     3 ⎞        ⎛                2     3 ⎞        ⎛      \n",
       "⎪⎜  142   32⋅x   7⋅x     x  ⎟        ⎜  32   8⋅x   2⋅x     x  ⎟        ⎜125   \n",
       "⎪⎜- ─── + ──── - ──── + ────⎟⋅c[5] + ⎜- ── + ─── - ──── + ────⎟⋅c[7] + ⎜─── - \n",
       "⎪⎜   3     Δ       2       3⎟        ⎜  3     Δ      2       3⎟        ⎜ 6    \n",
       "⎪⎝                Δ     2⋅Δ ⎠        ⎝              Δ     6⋅Δ ⎠        ⎝      \n",
       "⎪                                                                             \n",
       "⎪                                                    ⎛                2     3 \n",
       "⎪                                                    ⎜256   32⋅x   4⋅x     x  \n",
       "⎪                                                    ⎜─── - ──── + ──── - ────\n",
       "⎪                                                    ⎜ 3     Δ       2       3\n",
       "⎪                                                    ⎝              Δ     6⋅Δ \n",
       "⎪                                                                             \n",
       "⎪      ⎛               2     3 ⎞        ⎛              2      3 ⎞        ⎛    \n",
       "⎪      ⎜  5   7⋅x   5⋅x     x  ⎟        ⎜  1    x     x      x  ⎟        ⎜2   \n",
       "⎪      ⎜- ─ + ─── - ──── + ────⎟⋅c[2] + ⎜- ─ + ─── - ──── + ────⎟⋅c[4] + ⎜─ - \n",
       "⎪      ⎜  6   2⋅Δ      2      3⎟        ⎜  6   2⋅Δ      2      3⎟        ⎜3   \n",
       "⎩      ⎝            2⋅Δ    2⋅Δ ⎠        ⎝            2⋅Δ    6⋅Δ ⎠        ⎝    \n",
       "\n",
       " ⎞        ⎛                2     3 ⎞                                          \n",
       " ⎟        ⎜218   42⋅x   8⋅x     x  ⎟                                          \n",
       "─⎟⋅c[5] + ⎜─── - ──── + ──── - ────⎟⋅c[7]                     for 5⋅Δ ≤ x ∧ x \n",
       "3⎟        ⎜ 3     Δ       2       3⎟                                          \n",
       " ⎠        ⎝              Δ     2⋅Δ ⎠                                          \n",
       "                                                                              \n",
       "          2     3 ⎞        ⎛               2     3 ⎞                          \n",
       " 8⋅x   2⋅x     x  ⎟        ⎜50   16⋅x   5⋅x     x  ⎟                          \n",
       " ─── + ──── - ────⎟⋅c[3] + ⎜── - ──── + ──── - ────⎟⋅c[5]     for 3⋅Δ ≤ x ∧ x \n",
       "  Δ      2       3⎟        ⎜3     Δ       2       3⎟                          \n",
       "        Δ     6⋅Δ ⎠        ⎝             Δ     2⋅Δ ⎠                          \n",
       "                                                                              \n",
       "           2     3 ⎞                                                          \n",
       " 49⋅x   7⋅x     x  ⎟                                                          \n",
       " ──── + ──── - ────⎟⋅c[6]                                     for 6⋅Δ ≤ x ∧ x \n",
       " 2⋅Δ       2      3⎟                                                          \n",
       "        2⋅Δ    6⋅Δ ⎠                                                          \n",
       "                                                                              \n",
       "            2      3 ⎞         3                                              \n",
       "1    x     x      x  ⎟        x ⋅c[3]                                         \n",
       "─ + ─── + ──── - ────⎟⋅c[2] + ───────                           for 0 ≤ x ∧ x \n",
       "6   2⋅Δ      2      3⎟             3                                          \n",
       "          2⋅Δ    2⋅Δ ⎠          6⋅Δ                                           \n",
       "                                                                              \n",
       "       2     3 ⎞        ⎛               2     3 ⎞                             \n",
       "x   3⋅x     x  ⎟        ⎜31   15⋅x   7⋅x     x  ⎟                             \n",
       "─ + ──── - ────⎟⋅c[2] + ⎜── - ──── + ──── - ────⎟⋅c[4]        for 2⋅Δ ≤ x ∧ x \n",
       "Δ      2      3⎟        ⎜6    2⋅Δ       2      3⎟                             \n",
       "    2⋅Δ    6⋅Δ ⎠        ⎝            2⋅Δ    2⋅Δ ⎠                             \n",
       "                                                                              \n",
       "      2     3 ⎞                                                               \n",
       "⋅x   x     x  ⎟                                                               \n",
       "── + ── + ────⎟⋅c[1]                                          for -2⋅Δ ≤ x ∧ x\n",
       "Δ     2      3⎟                                                               \n",
       "     Δ    6⋅Δ ⎠                                                               \n",
       "                                                                              \n",
       "    ⎛            2      3 ⎞                                                   \n",
       "    ⎜1    x     x      x  ⎟                                                   \n",
       "] + ⎜─ + ─── + ──── + ────⎟⋅c[2]                               for -Δ ≤ x ∧ x \n",
       "    ⎜6   2⋅Δ      2      3⎟                                                   \n",
       "    ⎝          2⋅Δ    6⋅Δ ⎠                                                   \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "⋅c[0]                                                        for -3⋅Δ ≤ x ∧ x \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "          2     3 ⎞        ⎛                 2     3 ⎞                        \n",
       "25⋅x   5⋅x     x  ⎟        ⎜229   55⋅x   13⋅x     x  ⎟                        \n",
       "──── + ──── - ────⎟⋅c[4] + ⎜─── - ──── + ───── - ────⎟⋅c[6]   for 4⋅Δ ≤ x ∧ x \n",
       "2⋅Δ       2      3⎟        ⎜ 6    2⋅Δ        2      3⎟                        \n",
       "       2⋅Δ    6⋅Δ ⎠        ⎝              2⋅Δ    2⋅Δ ⎠                        \n",
       "                                                                              \n",
       "⎞                                                                             \n",
       "⎟                                                                             \n",
       "⎟⋅c[7]                                                        for 7⋅Δ ≤ x ∧ x \n",
       "⎟                                                                             \n",
       "⎠                                                                             \n",
       "                                                                              \n",
       "         2     3 ⎞        ⎛           2     3 ⎞                               \n",
       "2⋅x   2⋅x     x  ⎟        ⎜4   2⋅x   x     x  ⎟                               \n",
       "─── + ──── - ────⎟⋅c[3] + ⎜─ - ─── + ── - ────⎟⋅c[1]           for Δ ≤ x ∧ x <\n",
       " Δ      2       3⎟        ⎜3    Δ     2      3⎟                               \n",
       "       Δ     2⋅Δ ⎠        ⎝          Δ    6⋅Δ ⎠                               \n",
       "\n",
       "      \n",
       "      \n",
       "< 6⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 4⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 7⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< Δ   \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 3⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       " < -Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 0   \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< -2⋅Δ\n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 5⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "< 8⋅Δ \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       "      \n",
       " 2⋅Δ  \n",
       "      \n",
       "      "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now create the spline from the basis functions\n",
    "jastrow_cond_map = transpose_interval_and_coefficients(jastrow_sym_basis)\n",
    "c = IndexedBase('c',shape=(nknots+3))\n",
    "#c = MatrixSymbol('c',nknots+3,1)\n",
    "jastrow_spline = recreate_piecewise(jastrow_cond_map, c)\n",
    "jastrow_spline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAlBAMAAABc9LEgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$- \\frac{c_{0}}{2 \\Delta} + \\frac{c_{2}}{2 \\Delta}$$"
      ],
      "text/plain": [
       "  c[0]   c[2]\n",
       "- ──── + ────\n",
       "  2⋅Δ    2⋅Δ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALgAAAAlBAMAAAAD5T5pAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIAHP0QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- A - \\frac{c_{0}}{2 \\Delta} + \\frac{c_{2}}{2 \\Delta} = 0$$"
      ],
      "text/plain": [
       "     c[0]   c[2]    \n",
       "-A - ──── + ──── = 0\n",
       "     2⋅Δ    2⋅Δ     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJQAAAASBAMAAABP++oiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$c_{0} = - 2 A \\Delta + c_{2}$$"
      ],
      "text/plain": [
       "c₀ = -2⋅A⋅Δ + c[2]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Boundary conditions at r = 0 with the cusp (first derivative) condition\n",
    "\n",
    "cusp_val = Symbol('A')\n",
    "# Evaluate spline derivative at 0\n",
    "du = diff(jastrow_spline,xs)\n",
    "du_zero = du.subs(xs, 0)\n",
    "display(du_zero)\n",
    "\n",
    "# Solve following equation\n",
    "display(Eq(du_zero-cusp_val,0))\n",
    "\n",
    "# solve doesn't seem to work with Indexed value, substitute something else\n",
    "c0 = Symbol('c0')\n",
    "soln = solve(du_zero.subs(c[0],c0) - cusp_val, c0)\n",
    "display(Eq(c0, soln[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAAqBAMAAADxBIKUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmOMxn6jAvfE7OgfjfNc3YwHjtzGrdloAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$u{\\left (x \\right )} = \\frac{c_{5}}{6} + \\frac{2 c_{6}}{3} + \\frac{c_{7}}{6}$$"
      ],
      "text/plain": [
       "       c[5]   2⋅c[6]   c[7]\n",
       "u(x) = ──── + ────── + ────\n",
       "        6       3       6  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAAArBAMAAADRb+oyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTymROLlPCzF+qZou7IR/AbbnMFIQSNc6AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{d}{d x} u{\\left (x \\right )} = - \\frac{c_{5}}{2 \\Delta} + \\frac{c_{7}}{2 \\Delta}$$"
      ],
      "text/plain": [
       "d            c[5]   c[7]\n",
       "──(u(x)) = - ──── + ────\n",
       "dx           2⋅Δ    2⋅Δ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQsAAAAvBAMAAAAGMfL1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{d^{2}}{d x^{2}}  u{\\left (x \\right )} = \\frac{1}{\\Delta^{2}} \\left(c_{5} - 2 c_{6} + c_{7}\\right)$$"
      ],
      "text/plain": [
       "  2                             \n",
       " d          c[5] - 2⋅c[6] + c[7]\n",
       "───(u(x)) = ────────────────────\n",
       "  2                   2         \n",
       "dx                   Δ          "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Boundary conditions at r=r_cut.  For smoothness the value and derivatives should be 0.\n",
    "# Add zero value and derivatives at r_cut\n",
    "eq_v = jastrow_spline.subs(xs, rcut)\n",
    "display(Eq(Symbol('u')(xs),eq_v))\n",
    "\n",
    "eq_dv = diff(jastrow_spline, xs).subs(xs,rcut)\n",
    "display(Eq(diff(Symbol('u')(xs),xs),eq_dv))\n",
    "\n",
    "eq_ddv = diff(jastrow_spline, xs, 2).subs(xs,rcut)\n",
    "display(Eq(diff(Symbol('u')(xs),xs,2),eq_ddv))\n",
    "\n",
    "#eq_d3v = diff(jastrow_spline, xs, 3).subs(xs, rcut)\n",
    "#display(eq_d3v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALkAAAAmBAMAAABqsyf5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left\\{\\left ( c_{7}, \\quad - \\frac{c_{7}}{2}, \\quad c_{7}\\right )\\right\\}$$"
      ],
      "text/plain": [
       "⎧⎛    -c₇     ⎞⎫\n",
       "⎨⎜c₇, ────, c₇⎟⎬\n",
       "⎩⎝     2      ⎠⎭"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solve for all these equal to zero\n",
    "c5 = Symbol('c5')\n",
    "c6 = Symbol('c6')\n",
    "c7 = Symbol('c7')\n",
    "subs_list = {c[5]:c5, c[6]:c6, c[7]:c7}\n",
    "linsolve([v.subs(subs_list) for v in [eq_v, eq_dv, eq_ddv]], [c5, c6, c7])\n",
    "# QMCPACK enforces these conditions by setting c5, c6, c7 (the last three coefficients) to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
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  "language_info": {
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    "name": "ipython",
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
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