Fast B-spline class

fastBSpline - A fast, lightweight class that implements
non-uniform B splines of any order

Matlab's spline functions are very general. This generality comes at
the price of speed. For large-scale applications, including model
fitting where some components of the model are defined in terms of
splines, such as generalized additive models, a faster solution is
desirable.

The fastBSpline class implements a lightweight set of B-spline
features, including evaluation, differentiation, and parameter fitting.
The hard work is done by C code, resulting in up to 10x acceleration
for evaluating splines and up to 50x acceleration when evaluating
of spline derivatives.

Nevertheless, fastBSplines are manipulated using an intuitive, high-
level object-oriented interface, thus allowing C-level performance
without the messiness. Use CompileMexFiles to compile the required
files. If mex files are not available, evaluation will be done in .m
code, so you may still use the code if you can't use a compiler for your
platform.

B splines are defined in terms of basis functions:

y(x) = sum_i B_i(x,knots)*weights_i

B (the basis) is defined in terms of knots, a non-decreasing sequence
of values. Each basis function is a piecewise polynomial of order
length(knots)-length(weights)-1. The most commonly used B-spline is
the cubic B-spline. In that case there are 4 more knots than there
are weights. Another commonly used B-spline is the linear B-spline,
whose basis function are shaped like tents, and whose application
results in piecewise linear interpolation.

The class offers two static functions to fit the weights of a spline:
lsqspline and pspline. It includes facilities for computing the basis
B and the derivatives of the spline at all points.

Constructor:

sp = fastBSpline(knots,weights);

Example use:

%Fit a noisy measurement with a smoothness-penalized spline (p-spline)
x = (0:.5:10)';
y = sin(x*pi*.41-.9)+randn(size(x))*.2;
knots = [0,0,0,0:.5:10,10,10,10];
%Notice there are as many knots as observations

%Because there are so many knots, this is an exact interpolant
sp1 = fastBSpline.lsqspline(knots,3,x,y);
%Fit penalized on the smoothness of the spline
sp2 = fastBSpline.pspline(knots,3,x,y,.7);

clf;
rg = -2:.005:12;
plot(x,y,'o',rg,sp1.evalAt(rg),rg,sp2.evalAt(rg));
legend('measured','interpolant','smoothed');

fastBSpline properties:
outOfRange - Determines how the spline is extrapolated outside the
range of the knots
knots - The knots of the spline (read only)
weights - The weights of the spline (read only)

fastBSpline Methods:
fastBSpline - Construct a B spline from weights at the knots
lsqspline - Construct a least-squares spline from noisy measurements
pspline - Construct a smoothness-penalized spline from noisy
measurements
evalAt - Evaluate a spline at the given points
getBasis - Get the values of the underlying basis at the given points
Btimesy - Evaluate the product getBasis(x)'*y
dx - Returns another fastBSpline object which computes the derivative of
the original spline

Disclaimer: fastBSpline is not meant to replace Matlab's spline functions;
it does not include any code from the Mathworks