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Iterative System Solvers, Power Methods

Iterative System Solvers, Power Methods, and the Inverse Power Method for Boundary

Value Problems.

1. Code and test Jacobi and Gauss-Sidel solvers for arbitrary diagonally dominant linear systems.

2. Compare performance/results with tridiagonal Gaussian elimination solver for the problem arising from

-y’’=f on (0,1) with y(0)=0=y(1). You may also want to use sparse storage and MATLAB’s built in ’\’ operator

as a third solver.

3. Code and test a power method with deflation program to find all (approximate) eigenvalues/eigenvectors of

an arbitrary symmetric nxn matrix.

For full points you must use your Gauss-Sidel solver, but most credit can be acheived via use of the built in ’\’

operator. This applies to the next problem as well.

4. Code and test an inverse power method with deflation program to find the first few eigenvalues and eigenfunctions

(eigenvectors) of -y’’ = l y on (0,1) with y(0)=0=y(1).

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5. To shorten the project, this item is an Extra/Optional/Final Project idea.

Code and test an inverse power method with deflation program to find the first few eigenvalues and eigenfunctions

(eigenvectors) of - D u = l u on W = H0, 1L

2 with u=0 on ¶W .

You will need a function that solves - D u = f on W = H0, 1L

2 with u=0 on ¶W T. est this with

f(x,y)=2p2 sin(p x)sin(p y )E. ither use a Gauss-Sidel solver you code, or use sparse storage for the block tridiagonal

matrix together with the ’\’ operator.

6. Another Extra/Optional/Final Project Idea: Repeat problem 5 on an irregular subregion of H0, 1L

2.

7. Another Extra/Optional/Final Project Idea: Write a Gaussian elimination solver for the block tridiagonal

system coming from - D u = f on W = H0, 1L

2 with u=0 on ¶W a,nalogous to your existing tridiagonal solver.

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