mjo/cone/symmetric_pd: new module for symmetric positive-definite matrices.

[sage.d.git] / mjo / orthogonal_polynomials.py

diff --git a/mjo/orthogonal_polynomials.py b/mjo/orthogonal_polynomials.py
index 0da947cd6f0bf21705f501ec4523868135ff658f..589aa80b63def36bc6503fbbea7dc18ab1f7c104 100644 (file)
--- 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`::
@@ -60,15 +64,35 @@ 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 xrange(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`::
 
         sage: eq = lambda k: bool(legendre_p(k,x) == legendre_P(k,x))
-        sage: all([eq(k) for k in range(0,20) ]) # long time
+        sage: all( eq(k) for k in xrange(20) ) # long time
         True
 
     We can evaluate the result of the zeroth polynomial::
@@ -172,6 +196,6 @@ def legendre_p(n, x, a = -1, b = 1):
 
     # From Abramowitz & Stegun, (22.3.2) with alpha = beta = 0.
     # Also massaged to support finite field elements.
-    P = sum([ c(m)*g(m) for m in range(0,n+1) ])/(2**n)
+    P = sum( c(m)*g(m) for m in xrange(n+1) )/(2**n)
 
     return P