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Unformatted text preview: Solutions for Homework 6 Numerical Linear Algebra 1 Fall 2010 Problem 6.1 Define the random matrix A R 50 10 via Udiag (1 , 10- 1 , . . . , 10- 9 ) V T where U R 50 50 and V R 10 10 are random orthogonal matrices. The singular values of A are therefore 1 , 10- 1 , . . . , 10- 9 and the condition number is ( A ) 2 10 9 . Let A k be the matrix consisting of the first k columns of A and let ( A k ) 2 ,F be the condi- tion number of A k using either the matrix 2-norm or the matrix Frobenius norm. Implement both Classical and Modified Gram-Schmidt and assess their relative loss of orthogonality over the various ranges of columns by evaluating for each k ( A k ), k I k- Q T k Q k k 2 , and k I k- V T k V k k 2 , where Q k is computed via classical Gram-Schmidt and V k is computed by modified Gram-Schmidt. Of course, you should examine these values for several samples of U and V . Does the loss of orthogonality for classical occur more rapidly and severely when compared to the modified algorithm? Is the loss of orthogonality for the modified algorithm proportional to ( A k ) M ? Solution: The following table gives the results for a particular sample of U and V . A more rigorous solution would compute means over several samples. The computations were run in double precision so 2- 16 ....
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- Fall '06