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Lecture25

# Lecture25 - Successive Over Relaxation(SOR Relaxation...

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Successive Over Relaxation (SOR) Relaxation method - - - + - = - - - + - = - - - + - = - - - + - = 44 new 3 43 new 2 42 new 1 41 4 old 4 new 4 33 old 4 34 new 2 32 new 1 31 3 old 3 new 3 22 old 4 24 old 3 23 new 1 21 2 old 2 new 2 11 old 4 14 old 3 13 old 2 12 1 old 1 new 1 )/a x a x a x a (b )x (1 x )/a x a x a x a (b )x (1 x )/a x a x a x a (b )x (1 x )/a x a x a x a (b )x (1 x λ 2 2 21 1 23 3 24 4 22 2 2 2 2 2 21 1 23 3 24 4 22 G-S method ( ) / SOR method (1 ) (1 ) ( ) / new new old old new old new old new old old x b a x a x a x a x x x x b a x a x a x a = - - - = - + = - + - - - 1 ...( ) new x obtained =

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SOR Iterations + = - = - + = 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, and λ = 1.2 Rearrange [ ] [ ] = - × + × + × - = = - × - × + × - = - = - + × + × - = 7 . 7 12 / ) 6 . 0 ( 5 80 2 . 1 0 ) 2 . 0 ( x 29 . 7 8 / ) 6 . 0 ( 6 45 2 . 1 0 ) 2 . 0 ( x 6 . 0 4 / ) 2 0 0 ( 2 . 1 0 ) 2 . 0 ( x 3 2 1 First iteration ( 29 Seidel - Gauss : S - G x 1 x x old i S G i i ; - + = -
SOR Iterations = + × + × - = = - × + × - = = - + × + - × - = 4685 . 8 12 / )) 017 . 4 ( 5 80 ( 2 . 1 ) 7 . 7 ( ) 2 . 0 ( x 6767 . 1 8 / )) 017 . 4 ( 6 45 ( 2 . 1 ) 29 . 7 ( ) 2 . 0 ( x 017 . 4 4 / ) 2 7 . 7 29 . 7 ( 2 . 1 ) 6 . 0 ( ) 2 . 0 ( x 3 2 1 Second iteration Third iteration = + × + × - = = - × + × - = = - + × + × - = 1264 . 7 12 / )) 6402 . 1 ( 5 80 ( 2 . 1 ) 4685 . 8 ( ) 2 . 0 ( x 9385 . 4 8 / )) 6402 . 1 ( 6 45 ( 2 . 1 ) 6767 . 1 ( ) 2 . 0 ( x 6402 . 1 4 / ) 2 4685 . 8 6767 . 1 ( 2 . 1 ) 017 . 4 ( ) 2 . 0 ( x 3 2 1 Converges slower !! (see MATLAB solutions) There is an optimal relaxation parameter

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Successive Over Relaxation Relaxation factor w (= λ )
» [A,b]=Example A = 4 -1 -1 6 8 0 -5 0 12 b = -2 45 80 » x0=[0 0 0]' x0 = 0 0 0 » tol=1.e-6 tol = 1.0000e-006 » w=1.2; x = SOR(A, b, x0, w, tol, 100); i x1 x2 x3 .... 1.0000 -0.6000 7.2900 7.7000 2.0000 4.0170 1.6767 8.4685 3.0000 1.6402 4.9385 7.1264 4.0000 2.6914 3.3400 7.9204 5.0000 2.2398 4.0661 7.5358 6.0000 2.4326 3.7474 7.7091 SOR with λ = 1.2 Converge Slower !!

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» [A,b]=Example2 A = 4 -1 -1 6 8 0 -5 0 12 b = -2 45 80 » x0=[0 0 0]' x0 = 0 0 0 » tol=1.e-6 tol = 1.0000e-006 » w = 1.5; x = SOR(A, b, x0, w, tol, 100); i x1 x2 x3 .... 1.0000 -0.7500 9.2812 9.5312 2.0000 6.6797 -3.7178 9.4092 3.0000 -1.9556 12.4964 4.0732 4.0000 6.4414 -5.0572 11.9893 5.0000 -1.3712 12.5087 3.1484 6.0000 5.8070 -4.3497 12.0552 SOR with SOR with λ = 1.5 = 1.5 Diverged !!
Nonlinear Systems Simultaneous nonlinear equations Example 0 x x x x f 0 x x x x f 0 x x x x f 0 x x x x f n 3 2 1 n n 3 2 1 3 n 3 2 1 2 n 3 2 1 1 = = = = ) , , , , ( ) , , , , ( ) , , , , ( ) , , , , ( 4 x x 2 x 5 8 x x 7 x 5 x x 2 7 x 4 x 2 x 2 3 2 1 2 3 2 2 2 1 3 2 2 2 1 = + - = - + = + +

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Two Nonlinear Functions Circle x 2 + y 2 = r 2 Ellipse ( x/a ) 2 + ( y/b ) 2 = 1 { f(x,y)=0 g(x,y)=0 Solution depends on initial guesses
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Lecture25 - Successive Over Relaxation(SOR Relaxation...

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