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Solving Systems of Linear Equations: Iterative Methods

Solving Systems of Linear Equations: Iterative Methods. Contents. Introduction Basic Idea Jacobi Method Gauss-Seidel Method Successive Over Relaxation (SOR) Summary. Introduction (1/2).

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Solving Systems of Linear Equations: Iterative Methods

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  1. Solving Systems of Linear Equations: Iterative Methods

  2. Contents • Introduction • Basic Idea • Jacobi Method • Gauss-Seidel Method • Successive Over Relaxation (SOR) • Summary

  3. Introduction (1/2) • If systems of linear equations are very large, the computational effort of direct methods is prohibitively expensive • Three common classical iterative techniques for linear systems • The Jacobi method • Gauss-Seidel method • Successive Over Relaxation (SOR) method • Matlab’s built-in functions

  4. Introduction (2/2) • For systems that have coefficient matrices with the appropriate structure – especially large, sparse systems (many coefficients whose value is zero) – iterative techniques may be preferable

  5. Basic Idea • Convert the system into the equivalent system • Generate a sequence of approximation , where

  6. Jacobi Method (1/5) • Consider the two-by-two system • Start with • Simultaneous updating • New values of the variables are not used until a new iteration step is begun

  7. Jacobi Method (2/5) • Con’t

  8. Jacobi Method (3/5) • Consider the three-by-three system • Start with

  9. Jacobi Method (4/5) • Matlab function for jacobi method

  10. y x Jacobi Method (5/5) • Discussion • A necessary and sufficient condition for the convergence of the Jacobi method • The magnitude of the largest eigenvalue of the iteration matrix C be less than 1

  11. Gauss-Seidel Method (1/5) • Consider the two-by-two system • Start with • Sequential updating • New values of the variables are updated immediately

  12. Gauss-Seidel Method (2/5) • Con’t

  13. Gauss-Seidel Method (3/5) • Consider the three-by-three system • Start with

  14. Gauss-Seidel Method (4/5) • Matlab function for gauss-seidel method

  15. Gauss-Seidel Method (5/5) • Discussion • The Gauss-Seidel method is sensitive to the form of the co-efficient matrix A • The Gauss-Seidel method typically converges more rapidly than the Jacobi method • The Gauss-Seidel method is more difficult to use for parallel computation

  16. Successive Over Relaxation (SOR) (1/5) • Introduce an additional parameter, ω, that may accelerate the convergence of the iterations

  17. Successive Over Relaxation (SOR) (2/5) • Consider the three-by-three system

  18. Successive Over Relaxation (SOR) (3/5) • Required number of iterations for different values of the relaxation parameter • Start with • Tolerance = 0.00001

  19. Successive Over Relaxation (SOR) (4/5) • Matlab function for SOR

  20. Successive Over Relaxation (SOR) (5/5) • Discussion • The SOR method can be derived by multiplying the decomposed system obtained from the Gauss-Seidel method by the relaxation parameter w • The iterative parameter w should always be chosen such that 0 < w < 2

  21. Summary • Gauss-seidel method • Jacobi method • SOR method

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