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Unformatted text preview: Homework 6 Numerical Linear Algebra 1 Fall 2010 Solutions will be posted on Monday, 11/22/10 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 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 ( 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....
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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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