From c649291bc271a68a94e172f1946abffb6e5d9fdd Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 18 Nov 2012 02:23:54 -0500
Subject: [PATCH] Fix legendre_p for finite field elements.

---
 mjo/orthogonal_polynomials.py | 41 +++++++++++++++++++++++++----------
 1 file changed, 30 insertions(+), 11 deletions(-)

diff --git a/mjo/orthogonal_polynomials.py b/mjo/orthogonal_polynomials.py
index 808a558..0da947c 100644
--- a/mjo/orthogonal_polynomials.py
+++ b/mjo/orthogonal_polynomials.py
@@ -39,7 +39,7 @@ def legendre_p(n, x, a = -1, b = 1):
 
     Reuse the variable from a polynomial ring::
         sage: P.<t> = QQ[]
-        sage: legendre_p(2,t).simplify_rational()
+        sage: legendre_p(2,t)
         3/2*t^2 - 1/2
 
     If ``x`` is a real number, the result should be as well::
@@ -58,6 +58,11 @@ def legendre_p(n, x, a = -1, b = 1):
         [-179  242]
         [-484  547]
 
+    And finite field elements::
+
+        sage: legendre_P(3, GF(11)(5))
+        8
+
     TESTS:
 
     We should agree with Maxima for all `n`::
@@ -133,17 +138,30 @@ def legendre_p(n, x, a = -1, b = 1):
         # same field/ring as `x`.
         return x.parent()(1)
 
-    # Even though we know a,b are real we use the symbolic ring. This
-    # lets us return pretty expressions where possible.
-    a = SR(a)
-    b = SR(b)
-    n = ZZ(n) # Ensure that 1/(2**n) is not integer division.
-    dn = 1/(2**n)
+    # Even though we know a,b are real we use a different ring. We
+    # prefer ZZ so that we can support division of finite field
+    # elements by (a-b). Eventually this should be supported for QQ as
+    # well, although it does not work at the moment. The preference of
+    # SR over RR is to return something attractive when e.g. a=pi.
+    if a in ZZ:
+        a = ZZ(a)
+    else:
+        a = SR(a)
+
+    if b in ZZ:
+        b = ZZ(b)
+    else:
+        b = SR(b)
+
+    # Ensure that (2**n) is an element of ZZ. This is used later --
+    # we can divide finite field elements by integers but we can't
+    # multiply them by rationals at the moment.
+    n = ZZ(n)
 
     def phi(t):
-        # This is an affine map from [a,b] into [-1,1] and so preserves
-        # orthogonality.
-        return (2 / (b-a))*t + 1 - (2*b)/(b-a)
+        # This is an affine map from [a,b] into [-1,1] and so
+        # preserves orthogonality.
+        return (a + b - 2*t)/(a - b)
 
     def c(m):
         return binomial(n,m)*binomial(n, n-m)
@@ -153,6 +171,7 @@ def legendre_p(n, x, a = -1, b = 1):
         return ( ((phi(x) - 1)**(n-m)) * (phi(x) + 1)**m )
 
     # From Abramowitz & Stegun, (22.3.2) with alpha = beta = 0.
-    P = dn * sum([ c(m)*g(m) for m in range(0,n+1) ])
+    # Also massaged to support finite field elements.
+    P = sum([ c(m)*g(m) for m in range(0,n+1) ])/(2**n)
 
     return P
-- 
2.44.2