Lecture 25 - Gauss-Seidel Method 1 1 12 2 13 3 14 4 11 2 2...

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Unformatted text preview: Gauss-Seidel Method 1 1 12 2 13 3 14 4 11 2 2 21 23 3 24 4 22 3 3 31 32 34 4 33 4 4 41 42 43 44 1 1 2 1 2 13 ( ) / ( ) / ( ) / ( ) / new old old old new old old new new new ne old ne w new new e w n w x b a x a x a x a x b a a x a x a x b a a a x a x b x x x x x a x a a a =--- =--- =--- =--- Differ from Jacobi method by sequential updating: use new x i immediately as they become available Gauss-Seidel Method 1 1 1 1 1 12 2 13 3 14 4 11 1 1 2 2 21 1 23 3 24 4 22 1 3 3 31 1 32 2 34 4 33 4 4 41 1 42 2 43 3 44 ( ) / ( ) / ( ) / ( ) / j j j j j j j j j j j j j j j j x b a x a x a x a x b a x a x a x a x b a x a x a x a x b a x a x a x a------ =--- =--- =--- =--- use new x i at j th iteration as soon as they become available i s j i 1 j i j i i a, x all for 100 x x x < - =- % Example: = +-- =-- = +- 12 x 6 x x 3 7 x x 4 x 2 10 x 2 x 2 x 5 3 2 1 3 2 1 3 2 1 + +- = +- = +- = 6 12 x 6 1 x 6 3 x 4 7 x 4 1 x 4 2 x 5 10 x 5 2 x 5 2 x old 2 old 1 new 3 old 3 old 1 new 2 old 3 old 2 new 1 Jacobi Gauss-Seidel + +- = +- = +- = 6 12 x 6 1 x 6 3 x 4 7 x 4 1 x 4 2 x 5 10 x 5 2 x 5 2 x new 2 new 1 new 3 old 3 new 1 new 2 old 3 old 2 new 1 Jacobi and Gauss-Seidel Gauss-Seidel Iteration + =- =- + = 12 / ) x 5 80 ( x 8 / ) x 6 45 ( x 4 / ) 2 x x ( x 1 3 1 2 3 2 1 Assume x 1 = x 2 = x 3 = 0 as an initial guess Rearrange: [ ] [ ] =- + = =- - =- =- + = 4583 . 6 12 / ) 5 . ( 5 80 x . 6 8 / ) 5 . ( 6 45 x 5 . 4 / ) 2 ( x 3 2 1 First iteration 1 2 3 1 2 1 3 4 2 6 8 45 5 12 80 x x x x x x x-- = - + =- + = Consider: Gauss-Seidel Method = + = =- = = + +- = 7561 . 7 12 / )) 6146 . 2 ( 5 80 ( x 6641 . 3 8 / )) 6146 . 2 ( 6 45 ( x 6146 . 2 4 / ) 4583 . 6 6 2 ( x 3 2 1 = + = =- = = + +- = 6479 . 7 12 / )) 3550 . 2 ( 5 80 ( x 8587 . 3 8 / )) 3550 . 2 ( 6 45 ( x 3550 . 2 4 / ) 7561 . 7 6641 . 3 2 ( x 3 2 1 = + = =- = = + +- = 6569 . 7 12 / )) 3767 . 2 ( 5 80 ( x 8425 . 3 8 / )) 3767 . 2 ( 6 45 ( x 3767 . 2 4 / ) 6479 . 7 8587 . 3 2 ( x 3 2 1 Second iteration Third iteration Fourth iteration 6562 . 7 x , 8438 . 3 x , 3750 . 2 x : th 7 6563 . 7 x , 8437 . 3 x , 3750 . 2 x : th 6 6562 . 7 x ,...
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Fall '09 term at University of Florida.

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Lecture 25 - Gauss-Seidel Method 1 1 12 2 13 3 14 4 11 2 2...

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