Linear Systems – Iterative methods Jacobi Method

Transcript:Linear Systems – Iterative methods Jacobi Method:
Linear Systems – Iterative methods Jacobi Method Gauss-Siedel Method 1 Iterative Methods Iterative methods can be expressed in the general form: x(k) =F(x(k-1)) where s s.t. F(s)=s is called a Fixed Point Hopefully: x(k)  s (solution of my problem) Will it converge? How rapidly? 2 Iterative Methods Stationary: x(k+1) =Gx(k)+c where G and c do not depend on iteration count (k) Non Stationary: x(k+1) =x(k)+ak p(k) where computation involves information that change at each iteration 3 Iterative – Stationary Jacobi In the i-th equation solve for the value of xi while assuming the other entries of x remain fixed: In matrix terms the method becomes: where D, L and U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of M 4 Iterative – Stationary Gauss-Seidel Like Jacobi, but now assume that previously computed results are used as soon as they are available: In matrix terms the method becomes: where D, L and U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of M 5 Iterative – Stationary Successive Overrelaxation (SOR) Devised by extrapolation applied to Gauss-Seidel in the form of weighted average: In matrix terms the method becomes: where D, L and U represent the diagonal, the strictly lower-trg and strictly upper-trg parts of M ω is chosen to increase convergence 6 7 Jacobi iteration 8 Gauss-Seidel (GS) iteration Gauss-Seidel Method An iterative method. Basic Procedure: Algebraically solve each linear equation for xi Assume an initial guess solution array Solve for each xi and repeat Use absolute relative approximate error after each iteration to check if error is within a pre-specified tolerance. 9 Gauss-Seidel Method Why? The Gauss-Seidel Method allows the user to control round-off error. Elimination methods such as Gaussian Elimination and LU Decomposition are prone to prone to round-off error. Also: If the physics of the problem are understood, a close initial guess can be made, decreasing the number of iterations needed. 10 Gauss-Seidel Method Algorithm A set of n equations and n unknowns: . . . . . . If: the diagonal elements are non-zero Rewrite each equation solving for the corresponding unknown ex: First equation, solve for x1 Second equation, solve for x2 11 Gauss-Seidel Method Algorithm Rewriting each equation From Equation 1 From equation 2 From equation n-1 From equation n 12 Gauss-Seidel Method Algorithm General Form of each equation 13 Gauss-Seidel Method Algorithm General Form for any row ‘i’