Application 7 Table of Contents
Description of Tasks
Task 1
Compare the results for the Jacobi iteration and Gaussian elimination on matrices that are diagonally dominant. Tabulate the CPU time necessary to complete the iteration to a given number of digits of accuracy. Tabulate results for successfully larger matrices. Is there an intersection for the curves?
Task 2
Repeat the previous task using the Gauss-Seidel algorithm.
Task 3
Implement the steepest descent method for solving linear systems of equations Ax = b. Write up an entry in your software manual.
Task 4
Try out your steepest descent method on Hilbert matrices of size 4, 8, 16, 32. Explain your results.
Task 5
Implement the Conjugate Gradient method. Document the algorithm in an entry in your software manual.
Task 6
Try out the conjugate gradient method from the previous task on Hilbert matrices of order 4, 8, 16, and 32. Describe the results you obtained.
Task 7
Do an internet search on the use of iterative methods for the solution of linear systems of equations. Provide a list of at least three methods that do not use preconditioning of the system.
Task 8
Look for internet sites that include descriptions of preconditioning of systems of equations. Document at least 3 different preconditioning strategies.
Task 9
Compare results on symmetric positive definite linear systems of equations using Jacobi versus the conjugate gradient methods. Disucss the results on systems with at least 500 equations and unknowns.
Task 10
Describe an algorithm for computing the solution of linear least squares systems using Jacobi iteration. Create an entry in your software manual for your algorihm and an example. You can use one of the examples in the current textbook.