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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Q T Q = I is an orthogonal matrix and is a diagonal matrix whose diagonal elements are positive and also are the eigenvalues of A . Dene x = Q T x and b = Q T b Ax = b and x = b Given x and x , dene 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 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...
View Full Document

This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.

Ask a homework question - tutors are online