Lecture+27--Relaxation+method+_+Nonlinear+system

# Lecture+27--Relaxation+method+_+Nonlinear+system -...

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Relaxation Method HIGH BIRD CHECKLIST • If you know the target is going to be high, keep your muzzles higher than normal at the ready position. • Do not aim at the target. If you do you’ll miss behind. High birds often appear to be travelling slower than they really are, tempting you to aim. You must remember to swing the gun - and keep the muzzles moving after you’ve pulled the trigger.

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Relaxation Method Relaxation (weighting) factor Gauss-Seidel method:  = 1 Over-relaxation 1 <  < 2 Under-relaxation 0 <  < 1 Successive Over-relaxation (SOR) (1 ) new n i i ew x x λ = + - old i x old i x new i x new i x ( ) new i new i x x + = - old old i i x x ( 1)( ) new i new new i i x x x = + - - old i x
SOR Iterations Assume x1 = x2 = x3 = 0, and  = 1.2 Example: First iteration + = - = - + = 12 / ) x 5 80 ( x 8 / ) x 6 45 ( x 4 / ) 2 x x ( x 1 3 1 2 3 2 1 [ ] [ ] = - × + × + × - = = - × - × + × - = - = - + × + × - = 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 ( 29 Seidel - Gauss : S - G x 1 x x old i S G i i ; λ - + = -

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SOR Iterations Second iteration Third iteration Converges slower !! (see MATLAB solutions) There is an optimal relaxation parameter = + × + × - = = - × + × - = = - + × + - × - = 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 = + × + × - = = - × + × - = = - + × + × - = 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
Successive Over Relaxation Relaxation factor w (= )

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» [A,b]=Example A = 4 -1 -1 6 8 0 -5 0 12 b = -2 45 80 » x0=[0 0 0]'
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Lecture+27--Relaxation+method+_+Nonlinear+system -...

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