Lecture-10 - 1 Lecture-10 Theorems 5.2 and 5.3 Algorithms...

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Unformatted text preview: 1 Lecture-10 Theorems 5.2 and 5.3 Algorithms 5.1, 5.2 Theorem 5.2 Let x be any starting point and suppose that the sequence { x k } is generated by the conjugate direction algorithm. Then 1 , , for - = = k i p r i T k K and x k is minimizer of over the set x b Ax x x T T- = 2 1 ) ( f { } } , , { | 1- + = k p p span x x x K (3) 2 1 , , for ) ~ (- = = k i p x r i T K { } } , , { | 1- + = k p p span x x x K ) ( ) ( 1 1-- + + + = k k p p x h s s f s K ) , , ( 1 1- = k s s s s K 1 , , , ) ( *- = = k i h i K s s 1 , , ) ( 1 1 * *- = = + + + -- k i p p p x i T k k K K s s f First show that a point minimizes over the set (3) if and only if x ~ f Let Where Since is strictly convex quadratic, it has a unique minimizer: ) ( s h Chain rule ( 29 1 , , ~- = = k i p x r i T K ) ( ) ( x r b Ax x =- = f r ( x ) is the residual Proof (3) Proof 1 1 1--- + = k k k k Ap r r a ) ( ) ( x r b Ax x =- = f k k k k Ap r r a + = + 1 k k k k p x x a + = + 1 Use induction: Prove true for k= 1: (A) From (A) 1 Ap r r a + = 1 ) ( p Ap r p r T T a + = 1 Ap p p r p r T T T a + = 1 = p r T k T k k T k k Ap p p r- = a Because 3 Proof 1 1 1--- + = k k k k Ap r r a 1 1 1 1 1 1 = + =------ k T k k k T k k T k Ap p r p r p a 1 1 1 = + =--- k T i k k T i k T i Ap p r p r p a 2 , ,- = k i K Assume true for k- 1 (A) From (A) induction Definition k T k k T k k Ap p p r- = a And Conjugacy Therefore 1 , , for - = = k i p r i T k K QED 1 , , for - = = k i p r i T k K True for P k- 1 How do we select conjugate directions Eigenvalues of A are mutually orthogonal and conjugate wrt to A. Gram-Schmidt process can be modified to produce conjugate directions instead of orthogonal vectors. Both approaches are expensive. 4 Basic Properties of the CG 1- +- = k k k k p r p b 1 1 1--- = k T k k T k k Ap p Ap r b Each direction is chosen to be a linear combination of the steepest descent direction and the previous direction.descent direction and the previous direction....
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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-10 - 1 Lecture-10 Theorems 5.2 and 5.3 Algorithms...

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