# [Scipy-svn] r5461 - branches/0.7.x/scipy/special

scipy-svn@scip... scipy-svn@scip...
Tue Jan 13 02:34:28 CST 2009

Author: stefan
Date: 2009-01-13 02:34:19 -0600 (Tue, 13 Jan 2009)
New Revision: 5461

Modified:
branches/0.7.x/scipy/special/orthogonal.py
Log:
Backport changes to orthogonal docs.

Modified: branches/0.7.x/scipy/special/orthogonal.py
===================================================================
--- branches/0.7.x/scipy/special/orthogonal.py	2009-01-13 08:32:53 UTC (rev 5460)
+++ branches/0.7.x/scipy/special/orthogonal.py	2009-01-13 08:34:19 UTC (rev 5461)
@@ -1,8 +1,3 @@
-#!/usr/bin/env python
-#
-# Author:  Travis Oliphant 2000
-# Updated Sep. 2003 (fixed bugs --- tested to be accurate)
-
"""
A collection of functions to find the weights and abscissas for
@@ -14,30 +9,44 @@

Many recursion relations for orthogonal polynomials are given:

-a1n f_n+1 (x) = (a2n + a3n x ) f_n (x) - a4n f_n-1 (x)
+.. math::

+    a1n f_{n+1} (x) = (a2n + a3n x ) f_n (x) - a4n f_{n-1} (x)
+
The recursion relation of interest is

-P_n+1 (x) = (x - A_n) P_n (x) - B_n P_n-1 (x)
+.. math::

-where P has a different normalization than f.
+    P_{n+1} (x) = (x - A_n) P_n (x) - B_n P_{n-1} (x)

+where :math:P has a different normalization than :math:f.
+
The coefficients can be found as:

-A_n = -a2n / a3n
+.. math::

-B_n = ( a4n / a3n sqrt(h_n-1 / h_n))**2
+    A_n = -a2n / a3n
+    B_n = ( a4n / a3n \\sqrt{h_n-1 / h_n})^2

-     where
-             h_n = int_a^b w(x) f_n(x)^2
+where
+
+.. math::
+
+    h_n = \\int_a^b w(x) f_n(x)^2
+
assume:
-P_0(x) = 1
-P_-1(x) == 0

+.. math::
+
+    P_0 (x) = 1
+    P_{-1} (x) == 0
+
For the mathematical background, see [golub.welsch-1969-mathcomp]_ and
[abramowitz.stegun-1965]_.

-Functions:
+Functions::

gen_roots_and_weights  -- Generic roots and weights.
j_roots                -- Jacobi
@@ -66,7 +75,11 @@
Mathematical Functions: with Formulas, Graphs, and Mathematical
Tables*. Gaithersburg, MD: National Bureau of Standards.
http://www.math.sfu.ca/~cbm/aands/
+
"""
+#
+# Author:  Travis Oliphant 2000
+# Updated Sep. 2003 (fixed bugs --- tested to be accurate)

# Scipy imports.
import numpy as np