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"<script async src=\"https://www.googletagmanager.com/gtag/js?id=UA-59152712-8\"></script>\n",
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" gtag('js', new Date());\n",
"\n",
" gtag('config', 'UA-59152712-8');\n",
"</script>\n",
"\n",
"# Creating the Finite-Difference Table for Centered First and Second Derivatives, from 2nd through 10th-Order Accuracy\n",
"\n",
"<!-- abstract? -->\n",
"\n",
"## *Courtesy Brandon Clark*"
]
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"text": [
"Installing astropy, needed for creating the output table. Please wait a few seconds...\n",
"\u001b[33mWARNING: Retrying (Retry(total=4, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2436baa30>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/pip/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=3, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2436bac40>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/pip/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=2, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2436b58b0>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/pip/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=1, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2436b5dc0>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/pip/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=0, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2436b5cd0>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/pip/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=4, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2429bdd90>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/astropy/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=3, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2429bdf40>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/astropy/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=2, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2429c9130>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/astropy/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=1, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2429c92e0>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/astropy/\u001b[0m\n",
"\u001b[33mWARNING: Retrying (Retry(total=0, connect=None, read=None, redirect=None, status=None)) after connection broken by 'NewConnectionError('<pip._vendor.urllib3.connection.HTTPSConnection object at 0x7ff2429c9490>: Failed to establish a new connection: [Errno -2] Name or service not known')': /simple/astropy/\u001b[0m\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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