Lecture-14

Lecture-14 - Lecture-14 Rate of Convergence of CG Algorithm...

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1 Lecture-14 Rate of Convergence of CG Algorithm 5.2 ) ( ; 1 ; ; ; ; ; 0 0 , , ; Given 1 1 1 1 1 1 1 1 0 0 0 0 0 while end k k p r p r r r r Ap r r p x x Ap p r r r While k r p b Ax r set x k k k k k T k k T k k k k k k k k k k k T k k T k k k + + - + + - - - + + + + + + + + b b a a a We only need to know values of x, p and r only for 2 iterations. Major computations: matrix-vector product, two inner products, and three vector sums.
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2 Key points • According to theorem 5.3 Algorithm 5.2 should converge at most n steps. • Convergence less than n iterations, depending on the eigenvalues of matrix A. • If A dose not have favorable eigenvalues, then precondition A to get faster convergence. Theorem 5.4 If A has only r distinct eigenvalues, then the CG iteration will terminate at the solution in at most r iterations.
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3 Main points [ ] 2 * 0 2 max 1 min 2 * 1 || || ) ( 1 || || A i k i n i P A k x x P x x k - + - + l l Show are also eigenvectors of i n ) ( A P k Want to show: { } { } 0 0 0 1 0 , , , span , , , span r A Ar r p p p k k K K = (Theorem 5.3) Use this: k k k A A I A P g g g + + + = K 1 0 * ) ( Define polynomial Use orthogonal eigenvectors of A. i n Rate of Convergence k k k A A I A P g g g + + + = K 1 0 * ) ( 0 * 0 1 ) ( r A P x x k k + = + Az z z T A = 2 || || ) ( ) ( || || 2 1 * 2 * x x x x A f f - = - k k k p p x x a a + + + = + K 0 0 0 1 0 0 1 0 0 0 1 r A Ar r x x k k k g g g + + + + = + K By construction { } { } 0 0 0 1 0 , , , span , , , span r A Ar r p p p k k K K = (Theorem 5.3) Define polynomial: Therefore Now x b Ax x x T T - = 2 1 ) ( f (D)
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4 Rate of Convergence Az z z T A = 2 || || ) ( ) ( ) 2 1 ( 2 1 2 1 2 1 2 1 2 1 2 1 2
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Lecture-14 - Lecture-14 Rate of Convergence of CG Algorithm...

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