Tutorial-Finite_Difference_Derivatives-FDtable_soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Creating the Finite-Difference Table for Centered First and Second  Derivatives, from 2nd through 12th-Order Accuracy\n",
    "\n",
    "## *Courtesy Brandon Clark*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Derivative Accuracy    -6       -5      -4     -3     -2    -1      0       1     2      3       4      5       6    \n",
      "---------- -------- -------- ------- ------- ------ ------ ---- ---------- ---- ------ ------ ------- ------ --------\n",
      "       1st        2                                        -1/2             1/2                                      \n",
      "       1st        4                                   1/12 -2/3             2/3  -1/12                               \n",
      "       1st        6                           -1/60   3/20 -3/4             3/4  -3/20   1/60                        \n",
      "       1st        8                    1/280 -4/105    1/5 -4/5             4/5   -1/5  4/105  -1/280                \n",
      "       1st       10          -1/1260   5/504  -5/84   5/21 -5/6             5/6  -5/21   5/84  -5/504 1/1260         \n",
      "       1st       12   1/5544  -1/385    1/56  -5/63  15/56 -6/7             6/7 -15/56   5/63   -1/56  1/385  -1/5544\n",
      "       2nd        2                                           1         -2    1                                      \n",
      "       2nd        4                                  -1/12  4/3       -5/2  4/3  -1/12                               \n",
      "       2nd        6                            1/90  -3/20  3/2     -49/18  3/2  -3/20   1/90                        \n",
      "       2nd        8                   -1/560  8/315   -1/5  8/5    -205/72  8/5   -1/5  8/315  -1/560                \n",
      "       2nd       10           1/3150 -5/1008  5/126  -5/21  5/3 -5269/1800  5/3  -5/21  5/126 -5/1008 1/3150         \n",
      "       2nd       12 -1/16632  2/1925  -1/112 10/189 -15/56 12/7 -5369/1800 12/7 -15/56 10/189  -1/112 2/1925 -1/16632\n"
     ]
    }
   ],
   "source": [
    "# Step 0: Import needed modules\n",
    "import numpy as np\n",
    "import finite_difference as fin\n",
    "from astropy.table import Table\n",
    "\n",
    "# Step 1: Set the maximum finite-difference accuracy order\n",
    "max_fdorder =12\n",
    "\n",
    "# Step 2: Set up table parameters\n",
    "#    One column for deriv order, one for deriv accuracy, and max_fdorder+1\n",
    "numcols = 2 + max_fdorder + 1\n",
    "#    8 rows: max_fdorder accuracy orders per derivative order, times 2 derivative orders (first & second derivative)\n",
    "numrows = int(max_fdorder/2 * 2)\n",
    "#    Center column index of table will be at 2 + max_fdorder/2  (zero-offset indexing)\n",
    "column_corresponding_to_zero_fd_point = 2 + int(max_fdorder/2)\n",
    "#    The table is initialized as a matrix of zeroes in numpy...\n",
    "numpy_matrix = np.zeros((numrows, numcols), dtype=object)\n",
    "#    Then we replace all elements with the empty string to match the Wikipedia article.\n",
    "for row in range(numrows):\n",
    "    for col in range(numcols):\n",
    "        numpy_matrix[row,col] = \"\"\n",
    "\n",
    "# Step 3: Construct the first-order derivative finite difference coefficients\n",
    "rowcount = 0\n",
    "for fdorder in range(2, max_fdorder+1, 2): # loop runs from 2 to max_fdorder inclusive, skipping odd orders.\n",
    "    numpy_matrix[rowcount, 0] = \"1st\"\n",
    "    numpy_matrix[rowcount, 1] = fdorder     \n",
    "    fdcoeffs, fdstencl = fin.compute_fdcoeffs_fdstencl(\"D0\", fdorder)\n",
    "    for i in range(fdorder):\n",
    "        numpy_matrix[rowcount, column_corresponding_to_zero_fd_point + fdstencl[i][0]] = fdcoeffs[i]\n",
    "    rowcount += 1\n",
    "\n",
    "# Step 4: Construct the second-order derivative finite difference coefficients\n",
    "for fdorder in range(2, max_fdorder+1, 2): # loop runs from 2 to max_fdorder inclusive, skipping odd orders.\n",
    "    numpy_matrix[rowcount, 0] = \"2nd\"\n",
    "    numpy_matrix[rowcount, 1] = fdorder\n",
    "    fdcoeffs, fdstencl = fin.compute_fdcoeffs_fdstencl(\"DD00\", fdorder)\n",
    "    for i in range(fdorder+1):\n",
    "        numpy_matrix[rowcount, column_corresponding_to_zero_fd_point + fdstencl[i][0]] = fdcoeffs[i]\n",
    "    rowcount += 1  \n",
    "\n",
    "# Step 5: Construct an astropy table from the numpy matrix with the following header info, and then print it:\n",
    "colnames = ['Derivative','Accuracy']\n",
    "for i in range(-int(max_fdorder/2),int(max_fdorder/2)+1):\n",
    "    colnames.append(str(i))\n",
    "table = Table(numpy_matrix, names=colnames)\n",
    "table.pprint(max_width=-1)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}