From ff5b1d1514a419bbf8140658778d1e69e89b2664 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 18 Nov 2012 17:53:21 -0500 Subject: [PATCH] Add an example of a least squares problem. --- mjo/orthogonal_polynomials.py | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/mjo/orthogonal_polynomials.py b/mjo/orthogonal_polynomials.py index 0da947c..e8e9210 100644 --- a/mjo/orthogonal_polynomials.py +++ b/mjo/orthogonal_polynomials.py @@ -63,6 +63,26 @@ def legendre_p(n, x, a = -1, b = 1): 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`:: -- 2.43.2