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# hw8 - Q T Q = I is an orthogonal matrix and Λ is a...

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Homework 8 Foundations of Computational Math 1 Fall 2010 The solutions will be posted on Friday, 11/19/10 Problem 8.1 Textbook, page 330, Problem 6 Problem 8.2 Consider solving a linear system Ax = b where A is symmetric positive definite using steepest descent. (8.2.a) Suppose you use steepest descent without preconditioning. Show that the resid- uals, r k and r k +1 are orthogonal for all k . (8.2.b) Suppose you use steepest descent with preconditioning. Are the residuals, r k and r k +1 are orthogonal for all k ? If not is there any vector from step k that is guaranteed to be orthogonal to r k +1 ? Problem 8.3 Let A = Q Λ Q T be a symmetric positive definite matrix where
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Unformatted text preview: Q T Q = I is an orthogonal matrix and Λ is a diagonal matrix whose diagonal elements are positive and also are the eigenvalues of A . Deﬁne ˜ x = Q T x and ˜ b = Q T b Ax = b and Λ˜ x = ˜ b Given x and ˜ x , deﬁne the sequence x k as the sequence of vectors produced by steepest descent applied to Ax = b and the sequence ˜ x k as the sequence of vectors produced by steepest descent applied to Λ˜ x = ˜ b . Let e k = x k-x and ˜ e k = ˜ x k-˜ x . Show that if ˜ x = Q T x then k e k k 2 = k ˜ e k k 2 , k > 1...
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