Tutorial-Finite_Difference_Derivatives-FDtable_soln.ipynb

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    "# Creating the Finite-Difference Table for Centered First and Second  Derivatives, from 2nd through 10th-Order Accuracy\n",
    "\n",
    "## *Courtesy Brandon Clark*"
   ]
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      "Installing astropy, needed for creating the output table. Please wait a few seconds...\n",
      "astropy installed.\n",
      "Derivative Accuracy    -5      -4     -3     -2   -1      0       1    2     3      4      5   \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",
      "       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"
     ]
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   "source": [
    "print(\"Installing astropy, needed for creating the output table. Please wait a few seconds...\")\n",
    "!pip install -U pip astropy > /dev/null\n",
    "print(\"astropy installed.\")\n",
    "# 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 computed in the table\n",
    "max_fdorder = 10\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)"
   ]
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