quiz3 and ans - AMSC/CMSC 660 Quiz 3 Fall 2006 1(10 Suppose...

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AMSC/CMSC 660 Quiz 3 , Fall 2006 1. (10) Suppose we have factored A = LU and now we need to solve a linear system ( A - ZV T ) x = b , where Z and V have dimension n × k and k is much less than n . Write Matlab code to do this accurately and efficiently. You might want to use the Sherman-Morrison-Woodbury formula ( A - ZV T ) - 1 = A - 1 + A - 1 Z ( I - V T A - 1 Z ) - 1 V T A - 1 . (Don’t use matrix inverses!) Answer: This is Exercise 14. We use several facts to get an algorithm that is O ( kn 2 ) instead of O ( n 3 ): x = ( A - ZV T ) - 1 b = ( A - 1 + A - 1 Z ( I - V T A - 1 Z ) - 1 V T A - 1 ) b . Forming A - 1 from LU takes O ( n 3 ) operations, but forming A - 1 b as U \ ( L \ b ) uses forward and backward substitution and just takes O ( n 2 ). ( I - V T A - 1 Z ) is only k × k , so factoring it is cheap: O ( k 3 ). Matrix multiplication is associative. The Matlab code is: y = U \ (L \ b); Zh = U \ (L \ Z); t = (eye(k) - V’*Zh) \ (V’*y); x = y + Zh*t; 1
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2. (10) Denote the SVD of the 2 × 2 matrix A by UΣV T . (a) Express the solution to the linear system Ax = b as x = α 1 v 1 + α 2 v 2 where V = [ v 1 , v 2 ].
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