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Unformatted text preview: CS 290N/219: Sparse matrix algorithms: Homework 4 Assigned October 28, 2009 Due by class Wednesday, November 4 1. [20 points] (a) Find a 2-by-2 matrix A that is symmetric and nonsingular, but for which neither A nor − A is positive definite. What are the eigenvalues of A ? Find a 2-vector y such that y T Ay < 0. (b) For A as above, find a 2-vector b such that the conjugate gradient algorithm, when started with the zero vector as an initial guess, does not converge to the solution of Ax = b . Show what happens on the first two iterations of CG, as in the October 28 class slides. How do you know it won’t converge to the right answer? 2. [40 points] In this problem you’ll actually prove that CG works in at most n steps, assuming that real numbers are represented exactly. (This is not a realistic assumption in floating-point arithmetic, or on any computer with a finite amount of hardware, but it gives a solid theoretical underpinning to CG.) Let A be an n-by- n symmetric, positive definite matrix, and let b be an n-vector. We start with the idea of searching through n-dimensional space for the value of x that minimizes f ( x ) = 1 2 x T Ax − b T x , which is the x that satisfies...
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This note was uploaded on 12/27/2011 for the course CMPSC 290h taught by Professor Chong during the Fall '09 term at UCSB.
- Fall '09