Lecture-15 - Lecture-15 Homework, Rate of Convergence of...

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1 Lecture-15 Homework, Rate of Convergence of CG, preconditioning, FR-GC, PR-GC Homework (Due April 17) • 5.1 • 5.9 • Proof for Theorem 5.5 (see the slides)
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2 Theorem 5.4 If A has only r distinct eigenvalues, then the CG iteration will terminate at the solution in at most r iterations. Theorem 5.5 2 * 0 2 1 1 2 * 1 || || || || A k n k n A k x x x x - + - - - - + l l l l If A has eigenvalues we have n l l l K 2 1 n k n k n l l l l , , , , , 1 1 K K + - - Eigenvalues
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3 Proof ) ( 1 ) ( 1 l l l k k P Q + = + Maximum value attained by Q on the remaining eigenvalues is precisely n k n k n l l l l , , , , , 1 1 K K + - - Eigenvalues + - - - 1 1 l l l l k n k n Q has roots at k largest eigenvalues As well as at mid point 1 1 , , , + - - k n n n l l l K k n - l l and 1 Select polynomial of degree k such that ) ( l k P [ ] 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 (C) 2 * 0 2 1 1 2 * 1 || || || || A k n k n A k x x x x - + - - - - + l l l l Homework: show this Proof ) )( ( ) )( ( ) 1 ( ) ( 1 2 1 1 2 1 1 1 + + + + - - - - - = k k k k k k Q t l t l t l t l t t t t l K K 1 ) 0 ( , , 1 for 0 ) ( 1 1 = + - = = + + k i k Q n k n i Q K l 1 ) ( 1 - + l k Q Is polynomial of degree k+ 1 with root at 0 = l l l ) 1 ) ( ( 1 - = + k k Q P Degree k 2 max 1 min )] ( 1 [ i k i n i P P k l l + 2 1 1 2 max 1 2 max 1 min )] ( 1 [ )] ( 1 [ 0 + - = + + - - l l l l l l l l k n k n i k i n i i k i n i P P P k (B) Define polynomial : Assume eigenvalues take k distinct values: and n k n l l , , , 1 K + - k t t t < < , , 2 1 K 2 1 1 l l t + = - + k n k
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Example } , , }{ , , , { 1 1 n m n m n l l l l K K + - - Clustered around 1 A A m x x x x || || || || * 0 * 1 - - + e 2 * 0 2 1 1 2 * 1 || || || || A k n k n A k x x x x - + - - - - + l l l l m largest eigenvalues For small value of CG will converge in only m+ 1 steps. e
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This note was uploaded on 06/12/2011 for the course COT 6505 taught by Professor Shah during the Spring '07 term at University of Central Florida.

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Lecture-15 - Lecture-15 Homework, Rate of Convergence of...

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