14 - 3/4/2010 Iterative Methods: Basic idea: Convert Ax = b...

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3/4/2010 1 Iterative Methods: Basic idea: Convert A x = b into x = C x + d Make an initial guess at the solution. This guess is x 0 . Apply x i+1 = C x i + d until solution is acceptable Conceptually similar to iterative root finding techniques (e.g. Newton’s Method). The difference is that x is now a vector of values. Advantages: Speed Roundoff errors do not accumulate Disadvantages: Might not work (solution might not converge) Conversion of equations: a b a b x a a a a a a a a a b b b x a a a a a a a a a b Ax / / / 1 / / / 1 : by equation each Divide : Start with 22 2 11 1 22 23 22 21 11 13 11 12 ii 3 2 1 33 32 31 23 22 21 13 12 11 b a b a b x a a a a a a a a x a b a b a b x a a a a a a a a a a a a x a b a a a a / / 0 / 0 / / / 0 1 0 0 0 1 0 0 0 1 : Rearrange / / / 0 / / / 0 / / / 0 1 0 0 0 1 0 0 0 1 : LHS up Break / 1 / / 22 2 11 1 22 23 22 21 11 13 11 12 33 3 22 2 11 1 33 32 33 31 22 23 22 21 11 13 11 12 33 3 33 32 33 31 ii i i ii ij ij ii a b d a a c c a b a b a b x a a a a a a a a a a a a x d Cx x a a a a a / , / , 0 : Line Bottom / / / 0 / / / 0 / / / 0 : have now We / / / 33 3 22 2 11 1 33 32 33 31 22 23 22 21 11 13 11 12 33 3 33 32 33 31
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3/4/2010 2 Jacobi: x= x0; while true xold = x; x = C * xold + d; if x and xold are close enough, break, end end Gauss –Seidel: x= x0 while true xold = x;
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14 - 3/4/2010 Iterative Methods: Basic idea: Convert Ax = b...

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