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Unformatted text preview: 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 2. (10) Denote the SVD of the 22....
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This homework help was uploaded on 02/05/2008 for the course CMSC 660 taught by Professor Oleary during the Fall '06 term at Maryland.
- Fall '06