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linear_algebra_hmwk4

# linear_algebra_hmwk4 - ACS I Linear Algebra Homework 4 Due...

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ACS I Linear Algebra Homework # 4 Due: Monday, 2/14 1. SOR Algorithm Write a code to implement the component form of the SOR method for a symmetric positive definite matrix. As a stopping criteria use bardbl vectorx k +1 vectorx k bardbl 2 bardbl vectorx k +1 bardbl 2 10 - 4 or a maximum number of steps of 500. Test your code for SOR on the linear system Avectorx = vector b . 4 6 8 2 6 10 16 6 8 16 36 20 2 6 20 15 1 2 3 4 = 48 98 228 134 using ω = 1, i.e., Gauss-Seidel. Use a starting point of vectorx 0 = vector 0. a. Verify that A is positive definite. b. Use your SOR code to solve the given linear system for ω = 0 . 1 , 0 . 2 , . . ., 1 . 9. Make a table outputting the following information: (i) ω , (ii) the number of iterations required to get to the given tolerance and (iii) ρ ( P ω ) where P ω is the iteration matrix for SOR for the given value of ω and ρ ( A ) denotes the spectral radius of the matrix A . (You can use Matlab to compute the spectral radius.) Plot ω . What is the optimal choice of ω for this problem? Relate this to ρ ( P ω ) and compare with theoretical results.

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