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Fourth Computing Project \\
MA375, Fall 1993 \\
Professor Coutsias \\
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This project is due November 13. 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 write a subroutine implementing Newton's method for solving
systems of nonlinear equations. You should employ a damping technique to
guarantee global convergence. The routine should accept as arguments:
\begin{description}
\item[(i)] User supplied subroutines for evaluating the function and its
Jacobian. 
\item[(ii)] User supplied initial guess.
\item[(iii)] User supplied error tolerance and upper bound on the number of
iterations. 
\end{description}
Use the lapack codes you obtained for the last assignment to do the
required linear algebra.

Try out your code on the following problems:
\begin{enumerate}
\item Find {\em all} points where the circle $x_{1}^{2}+x_{2}^{2}=1$ and the
ellipse $4x_{1}^{2}+ \frac {x_{2}^{2}}{16} =1$ intersect.
\item Find as many solutions as you can for:
\begin{displaymath}
x_{i+1}-2x_{i}+x_{i-1}+.002e^{x_{i}} = 0, \ \ i=1, \ldots ,9; \ \
(x_{0}=x_{10}=0.)
\end{displaymath}
\item Minimize: 
\begin{displaymath}
g(x_{1},x_{2},x_{3}) = \frac {x_{1}+5x_{2}}{3x_{1}^{2}+6x_{2}^{2}+x_{3}^{2}+10}
 .
\end{displaymath}
\item Minimize:
\begin{displaymath}
w(x_{1},x_{2},x_{3}) = (1+x_{3}^{4})e^{2x_{1}^{2}+3x_{2}^{2}} .
\end{displaymath}
\end{enumerate}

In all cases display your initial guess, the number of iterations, as well as
the final solution to which you converge. After you have the converged result,
$x^{\ast}$, go back and compute $\| e^{k} \|= \|x^{k}-x^{\ast}\|$. Study the
behavior of $\frac  {\| e^{k+1} \|}{\| e^{k} \|^{2}}$. Do you observe
quadratic convergence? If not, try to explain why not. 

Your programs should be well written and well documented. Do NOT
turn in any listings of library programs, but do email to me 
(hagstrom@mimir.unm.edu) your subroutine and 
drivers. A well-written description of all
your results should be included.


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