X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Forthogonal_polynomials.py;h=7e094fc201076938120972f268b55fd628470a47;hb=2fa2a7a2f6a5d5f1628a4f5cf3301d5e7f670038;hp=0da947cd6f0bf21705f501ec4523868135ff658f;hpb=c649291bc271a68a94e172f1946abffb6e5d9fdd;p=sage.d.git

diff --git a/mjo/orthogonal_polynomials.py b/mjo/orthogonal_polynomials.py
index 0da947c..7e094fc 100644
--- a/mjo/orthogonal_polynomials.py
+++ b/mjo/orthogonal_polynomials.py
@@ -28,6 +28,10 @@ def legendre_p(n, x, a = -1, b = 1):
     returned. Otherwise, the value of the ``n``th polynomial at ``x``
     will be returned.
 
+    SETUP::
+
+        sage: from mjo.orthogonal_polynomials import legendre_p
+
     EXAMPLES:
 
     Create the standard Legendre polynomials in `x`::
@@ -60,9 +64,29 @@ def legendre_p(n, x, a = -1, b = 1):
 
     And finite field elements::
 
-        sage: legendre_P(3, GF(11)(5))
+        sage: legendre_p(3, GF(11)(5))
         8
 
+    Solve a simple least squares problem over `[-\pi, \pi]`::
+
+        sage: a = -pi
+        sage: b = pi
+        sage: def inner_product(v1, v2):
+        ...       return integrate(v1*v2, x, a, b)
+        ...
+        sage: def norm(v):
+        ...       return sqrt(inner_product(v,v))
+        ...
+        sage: def project(basis, v):
+        ...       return sum([ inner_product(v, b)*b/norm(b)**2
+        ...                    for b in basis])
+        ...
+        sage: f = sin(x)
+        sage: legendre_basis = [ legendre_p(k, x, a, b) for k in range(0,4) ]
+        sage: proj = project(legendre_basis, f)
+        sage: proj.simplify_trig()
+        5/2*(7*(pi^2 - 15)*x^3 - 3*(pi^4 - 21*pi^2)*x)/pi^6
+
     TESTS:
 
     We should agree with Maxima for all `n`::