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Third Computing Project \\
Math/CS 375, Fall 1997 \\
Professor Coutsias \\
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This project is due October 28. Plan your work
so that if your computer is down a short period, your project will not be late.

This project will count as part of your grade so you must work independently.
It is permissible to discuss the project with your instructor, fellow students,
and friends. However, the computer programs and report must be done only
by the student doing the project.

The project is to compare various polynomial and piecewise polynomial
interpolants. In particular, you are to compare polynomial
interpolants computed using 9 and 17 equally spaced and Chebyshev nodes
and a cubic spline interpolant
using 9 and 17 equally spaced nodes. The functions
to be approximated are:
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\item[(i.)] $\cos{(\pi x)} \cdot e^{-x} , \ \ 0 \leq x \leq 4.$
\item[(ii.)] $\frac {1}{1+x^{2}} , \ \ -5 \leq x \leq 5.$
\item[(iii.)] $\sqrt{x} , \ \ 0 \leq x \leq 2.$
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For all 18 approximations plot the approximation with the true function and
also plot the error. Comment on which functions were most easily approximated
and which methods of approximation worked best. Give some theoretical
explanation for the observed results. (Precise error estimates are 
unnecessary.) For the polynomial 
interpolation, you may either use an exisitng routine (for example, Matlab's
interp) or write your own divided difference algorithm and
evaluation routine using nested multiplication. For the splines you are
to use the book's routines Spcoeff, Svalue.

Grading guidelines: Correct use of the interpolation functions or correct
programs if you write them 50\%; Completion of the examples 35\%, 
Discussion 15\%.

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