Math 543 Computer Project 02
For this project you will need to use a mathematics software tool package like
Matlab, Maple or Mathematica, or a programming language. The project will
involve the use of the four test functions
 exp(x), for x in [1,1],
 1/(1+25x^2), for x in [1,1],
 log(x), for x in [1,e], and
 exp(x^2/2)/sqrt(2 pi), for x in [0,5].

Construct a procedure or function that will
determine the coefficients for the cubic Bspline respresentation
for the clamped (or complete) interpolating spline S(x) for a set of
equally spaced data values.
The procedure should have as input parameters:
 m  the number of data points,
 f  an array of length m of function values,
 fp_1, fp_m  the function derivative values at the endpoints for the
domain of the function.
The output should be an array of length m+2 of cubic Bspline coefficients.

Construct a procedure or function that will evaluate the Bspline
representation for a clamped (or complete) interpolating cubic spline S(x)
at a particular point xp.
The procedure should have as input parameters:
 m  the number of data points,
 a  the left endpoint for the interpolation interval,
 b  the right endpoint for the interpolation interval,
 c  an array of length m+2 of cubic Bspline coefficients.
 xp  the spline evaluation point.
The output should be the value of the spline at xp.

Construct a procedure or function that will compute the integral of the
the Bspline representation for a clamped (or complete) interpolating cubic
spline S(x) over the interval of definition [a,b] for the original function
f(x) which S(x) interpolates.
The procedure should have as input parameters:
 m  the number of data points,
 a  the left endpoint for the interpolation interval,
 b  the right endpoint for the interpolation interval,
 c  an array of length m+2 of cubic Bspline coefficients.
The output should be the value of the integral of the spline.
 For each of the four functions determine coefficients for the spline
S(x) for m = 5, 9, 17 and 33, and
 determine the maximum absolute error for each m using 129 equally spaced
xp values;
 plot the error e(x) = f(x)  S(x) for m = 33;
 determine the integral of the spline for each m.
 Hand in your computer source code, graphs, and a brief discussion of
your results. Make sure that your source code is wellstructured and
reasonably efficient, with some comments to explain what you have done.