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

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

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