# hw5 - Due October 2 2008 CS 257(Luke Olson Homework#5...

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Due: October 2, 2008 CS 257 (Luke Olson): Homework #5 Problem 1 Problem 1 a. If A is a non-singular square matrix, prove that A T A is symmetric and positive deﬁnite. You can assume that all values in A are real. b. Experiment with the code below. What can you hypothesize about the condition number of A T A compared to A ? Listing 1: Code for problem 1b 1 n=5; 2 3 A = rand (n,n); 4 B = A’ * A; 5 fprintf ( ’cond(A) cond(A)ˆ2 cond(AˆT * A)\n’ ) 6 fprintf ( ’%g %g %g\n’ , cond (A), cond (A)ˆ2, cond (B)) c. Suppose you’re working on an embedded platform with no pre-built numerical libraries, and you need to solve Ax = b . However, you also have free access to A T A and A T b because they were computed previously for some unrelated reason in your code. You now have a choice of writing Cholesky factorization to solve A T Ax = A T b or Gaussian Elimination with scaled partial pivoting to solve Ax = b .

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hw5 - Due October 2 2008 CS 257(Luke Olson Homework#5...

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