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hw6 - Homework 6 Numerical Linear Algebra 1 Fall 2010...

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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 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 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 κ ( A k ), I k - Q T k Q k 2 , and I k - V T k V 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

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hw6 - Homework 6 Numerical Linear Algebra 1 Fall 2010...

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