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Lecture23

# Lecture23 - Jacobi Method x 1 new new x2 new x3 new x4 = b1...

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Jacobi Method - - - = - - - = - - - = - - - = 44 old 3 43 old 2 42 old 1 41 4 new 4 33 old 4 34 old 2 32 old 1 31 3 new 3 22 old 4 24 old 3 23 old 1 21 2 new 2 11 old 4 14 old 3 13 old 2 12 1 new 1 a x a x a x a b x a x a x a x a b x a x a x a x a b x a x a x a x a b x / ) ( / ) ( / ) ( / ) ( It is important to have diagonal dominance: 1, | | | | n ii ij j j i a a =

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- - - - - - - - = 9 3 2 3 3 12 6 1 5 2 10 1 3 2 1 8 A is strictly diagonally dominant because diagonal elements are greater than sum of absolute value of other elements in row is not diagonally dominant Diagonally Dominant Matrix - - - - - - - = 9 3 2 3 3 12 6 1 5 2 6 1 3 2 1 8 A 6<|-1|+2+5=8
Example Not diagonally dominant !! Order of the equation can be important Rearrange the equations to ensure convergence - - - = + - = - - = + - 0 8 6 1 1 4 12 0 5 45 x 8 x 6 2 x x x 4 80 x 12 x 5 2 1 3 2 1 3 1 - - - = + - = + - = - - 12 0 5 0 8 6 1 1 4 80 x 12 x 5 45 x 8 x 6 2 x x x 4 3 1 2 1 3 2 1

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* Achieving diagonal dominance through pivoting (before starting any iterative methods): Iterative Methods 1 2 3 6 2 1 11 2 2 5 1 2 1 7 5 x x x -     - =     -   1 2 3 6 2 1 11 1 1 2 5 5 x x x -     = -      
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• Spring '09
• RAPHAELHAFTKA
• 1 j, Gauss–Seidel method, Diagonally dominant matrix, 0.5000 6.0000 2.0000 2.6146 3.6641 3.0000 2.3550 3.8587 4.0000 2.3767 3.8425 5.0000 2.3749 3.8439 6.0000 2.3750 3.8437 7.0000 2.3750 3.8438 8.0000 2.3750 3.8437 Gauss, 1.1809 Gauss, 2.6146 Second

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