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{\large {\bf Computer Assignment 3} (due Wednesday, March 3)}
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\underline{\bf Jacobi method and Gauss-Seidel method}
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\noindent
{\bf Problem 1}
\noindent
Consider following systems of equations:
$$3x_1 + x_2 + x_3 = 5$$
$$x_1 + 7x_2 + 3x_3 = 11$$
$$2x_1 + 4x_3 = 5$$
and
$$x_1 + 5x_2 + x_3 = 7$$
$$9x_1 + 3x_2 + 3x_3 = 15$$
$$2x_1 + x_2 + 4x_3 = 7$$
\noindent
1) Compute necessary matrices, make norm estimates and find whether Jacobi and
Gauss-Seidel methods converge for these problems. If they do, estimate
analytically the number of iterations needed to find the answers with relative
errors less than 0.0001.
\noindent
If the estimates show that a method would not converge, reformulate the
problem so, that it would converge, and repeat the estimates.
\noindent
2) Write and execute programs for the Jacobi itertion and the Gauss-Seidel
iteration.
Use these programs to find the solutions of both problems above
(using both techniques!) with required precision.
\noindent
3) Find the required number of iterations in all cases. Compare it with the
analytical estimates. Compare the number of iterations required in different
techniques. Comment on the results. Use graphics to illustrate the
actual rate of convergence for both methods (for instance
plot $\|x^{(n)}-x^{(n-1)}\|/\|x^{(n)}\|$ for $n=1,2,\dots$).
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\noindent
{\bf Problem 2}
\noindent
Write and execute a program to find out if an $n \times n$ matrix A
is well- or ill-conditioned and apply this program to the matrices of
the problems above. Are the systems above well-conditioned?
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\noindent
{\bf Note:} Your programs should be written such that they can also
handle other examples, not only the particular examples given above!
\newpage
\centerline{\bf Format for Computer Assignments}
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Your task in each of the programming assignments is to write a brief paper
which answers the given questions and illustrates your ideas in clear and
concise prose. The report should separate the required tasks and document
each in the appropriate section: Analysis, Computer Program, Results.
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\noindent
{\bf Analysis (30\%):} Mathematical derivations necessary to solve the problem.
Brief description of all algorithms you plan to use in your code.
Discussion of numerical considerations (if applicable)
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\noindent
{\bf Computer Program (30\%):} The source code should be readable and printed
with margins. Internal comments should describe algorithms and variables,
relating them to those described in your Analysis section. Briefly describe
input and output to and from your code. Do not expect bugs to be found
during the grading process, rather use TA office hours for the assistance.
\bigskip
\noindent
{\bf Results (30\%):} Output of your program and explanation of the results.
Answers on qualitative questions. Discussion (why it worked, why it did not
work, comparison to the predictions, error bounds)
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\noindent
{\bf Style (10\%):} Logical, clear, concise, well documented report.
\bigskip
\noindent
Return the computer assignment as paper outprint to me at the beginning of
the class on the day due, {\bf and} send it also by e-mail to the TA
Darryl Whitlow (whitlow@math.ucdavis.edu)!!!
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