mws_gen_sle_ppt_seidel - Gauss-Siedel Method Major: All...

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1/10/2010 http://numericalmethods.eng.usf.edu 1 Gauss-Siedel Method Major: All Engineering Majors Authors: Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates
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Gauss-Seidel Method http://numericalmethods.eng.usf.edu
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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.
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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.
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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
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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
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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
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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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Gauss-Seidel Method Solve for the unknowns Assume an initial guess for [X] n - n 2 x x x x 1 1 Use rewritten equations to solve for each value of x i . Important: Remember to use the
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_sle_ppt_seidel - Gauss-Siedel Method Major: All...

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