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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 κ ( A ) 2 ≈ 10 9 . Let A k be the matrix consisting of the first k columns of A and let κ ( A k ) 2 ,F be the condi tion number of A k using either the matrix 2norm or the matrix Frobenius norm. Implement both Classical and Modified GramSchmidt and assess their relative loss of orthogonality over the various ranges of columns by evaluating for each k κ ( 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 GramSchmidt and V k is computed by modified GramSchmidt. 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 κ ( 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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This note was uploaded on 07/21/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.
 Fall '06
 gallivan

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