# hw5 - r k 6 = 0, do (a) k =-r T k p k p T k Ap k , (b) x k...

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Homework 5 – Math 104B, Winter 2011 Due on Thursday, February 24th, 2011 Section 8.1 4, 12, 13a, 13b. Section 8.2 1e, 7b, 8b, 11, 12c. Additional problem 1: Let A R n × n be a symmetric, positive deﬁnite matrix, and { p 0 , p 1 , . . . , p n - 1 } a set of A -conjugate directions. Consider the Conjugate Direction method seen in class for the solution of Ax = b : 1. Given x 0 , deﬁne r 0 = Ax - b . 2. For each k 0, (a) Deﬁne α k = - r T k p k p T k Ap k , (b) and let x k +1 = x k + α k p k . Prove that r T k p i = 0 , i = 0 , 1 , . . . , k - 1 , and that x k is the minimizer of φ ( x ) = 1 2 x T Ax - x T b over the set x 0 + span { p 0 , p 1 , . . . , p k - 1 } . Additional problem 2: Let A R n × n be a symmetric, positive deﬁnite matrix, and { p 0 , p 1 , . . . , p n - 1 } a set of A -conjugate directions. Consider the Conjugate Gradient method seen in class for the solution of Ax = b : 1. Given x 0 , deﬁne r 0 = Ax - b , and p 0 = - r 0 . 2. While

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Unformatted text preview: r k 6 = 0, do (a) k =-r T k p k p T k Ap k , (b) x k +1 = x k + k p k , (c) r k +1 = r k + k Ap k , (d) k = r T k +1 Ap k p T k Ap k , (e) p k +1 =-r k +1 + k +1 p k . (f) k k + 1. 1 2 Suppose that the k-th iterate generated by the algorithm is not the solution point x * . Prove that r T k r i = 0 , i = 0 , 1 , . . . , k-1 span { r , r 1 , . . . , r k } = span r , Ar , . . . , A k r , span { p , p 1 , . . . , p k } = span r , Ar , . . . , A k r , p T k Ap i = 0 , i = 0 , 1 , . . . , k-1 . Therefore the sequence { x k } converges to x * in at most n steps....
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## This note was uploaded on 12/27/2011 for the course MATH 104b taught by Professor Ceniceros,h during the Fall '08 term at UCSB.

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hw5 - r k 6 = 0, do (a) k =-r T k p k p T k Ap k , (b) x k...

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