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First Computing Project \\
MA/CS 375, Fall 1995 \\
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This project is due September 25. 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.

Your project is to prepare a Mathematica notebook on the computation of
zeros. Your basic algorithm should be the secant method. Its inputs should
be the function whose root is to be found, an initial guess, and a tolerance.
(See the class notebooks for a model of how this can be done. Choose $x_{-1}$
automatically to be some point near $x_{0}$.) Features of 
the algorithm should include a damping procedure to enhance global convergence
and a reliable convergence test. Examples should include:
\begin{description}
\item[i.] A computation of the first ten positive zeros of the order zero 
Bessel function, $J_{0}(x)$ with a relative error of no more than $10^{-6}$. 
First plot the function and estimate the zeros by
eye to generate initial approximations. Verify the superlinear convergence
of the method.
(The order zero Bessel function may be thought of
as a generalization of trigonometric functions to polar geometry. In
Mathematica, Bessel functions $J_{n}(x)$ are called $BesselJ[n,x]$. To find
out more about special functions, open the function browser on the help
menu. Under mathematical functions, hypergeometric related, you'll find the
complete family of Bessel functions and its relatives.) 
\item[ii.] Verify that your damping procedure is effective by finding the zero
of $\tanh{x}$ with initial approximations, $x_{0}=1,2,\ldots,10$. Verify that
your convergence test is good by comparing the accuracy of computed roots
to $\tanh{x}$, $100 \cdot \tanh{x}$, and $.01 \cdot \tanh{x}$.
\item[iii.] Use your method to compute a root of $x-\sin{x}$. Verify that
you obtain the desired accuracy and that Mathematica's FindRoot does not!.
Do you observe superlinear convergence? 

\end{description}

Grading of the assignment will roughly follow these guidelines: Correctness
of the secant code $20\%$, correct implementation of damping $15\%$, 
correct convergence test $15\%$, correct testing of convergence rate $15\%$,
completion of the Bessel example $10\%$, completion of the $\tanh{x}$
example $10\%$, completion of the $x-\sin{x}$ example $10\%$,
readability of the report $5\%$. 


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