nbm_ind_sle_ppt_seidel - http/numericalmethods.eng.usf.edu...

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Unformatted text preview: 08/11/11 http://numericalmethods.eng.usf.edu 1 Gauss-Siedel Method Industrial Engineering Majors Authors: Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Gauss-Seidel Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu Gauss-Seidel Method An iterative method. Basic Procedure :-Algebraically solve each linear equation for x i - Assume an initial guess solution array-Solve for each x i and repeat- Use absolute relative approximate error after each iteration to check if error is within a pre-specified tolerance. http://numericalmethods.eng.usf.edu 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. http://numericalmethods.eng.usf.edu Gauss-Seidel Method Algorithm A set of n equations and n unknowns: 1 1 3 13 2 12 1 11 ... b x a x a x a x a n n = + + + + 2 3 23 2 22 1 21 ... b x a x a x a x a n 2n = + + + + n n nn n n n b x a x a x a x a = + + + + ... 3 3 2 2 1 1 . . . . . . If: the diagonal elements are non-zero Rewrite each equation solving for the corresponding unknown ex: First equation, solve for x 1 Second equation, solve for x 2 http://numericalmethods.eng.usf.edu Gauss-Seidel Method Algorithm Rewriting each equation 11 1 3 13 2 12 1 1 a x a x a x a c x n n--- = nn n n n n n n n n n n n n n n n n n n n n n a x a x a x a c x a x a x a x a x a c x a x a x a x a c x 1 1 , 2 2 1 1 1 , 1 , 1 2 2 , 1 2 2 , 1 1 1 , 1 1 1 22 2 3 23 1 21 2 2---------------- =---- =--- = From Equation 1 From equation 2 From equation n-1 From equation n http://numericalmethods.eng.usf.edu Gauss-Seidel Method Algorithm General Form of each equation 11 1 1 1 1 1 a x a c x n j j j j ∑ ≠ =- = 22 2 1 2 2 2 a x a c x j n j j j ∑ ≠ =- = 1 , 1 1 1 , 1 1 1--- ≠ =--- ∑- = n n n n j j j j n n n a x a c x nn n n j j j nj n n a x a c x ∑ ≠ =- = 1 http://numericalmethods.eng.usf.edu Gauss-Seidel Method Algorithm General Form for any row ‘i’ . , , 2 , 1 , 1 n i a x a c x ii n i j j j ij i i =- = ∑ ≠ = How or where can this equation be used?...
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This note was uploaded on 08/11/2011 for the course MATH 663 taught by Professor Ron during the Spring '11 term at University of the Philippines Diliman.

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nbm_ind_sle_ppt_seidel - http/numericalmethods.eng.usf.edu...

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