Unformatted text preview: O ( kn 2 ) instead of O ( n 3 ) for dense matrices: • x = ( A − ZV T ) − 1 b = ( A − 1 + A − 1 Z ( I − V T A − 1 Z ) − 1 V T A − 1 ) b . • ²orming A − 1 from the LU decomposition takes O ( n 3 ) operations, but forming A − 1 b as U \ ( L \ b ) uses forward and backsubstitution and just takes O ( n 2 ). • ( I − V T A − 1 Z ) is only k × k , so factoring it is cheap: O ( k 3 ). (²orming it is more expensive, with a cost that is O ( kn 2 ).) • Matrix multiplication is associative. Using MATLAB notation, once we have formed [L,U]=lu(A) , the resulting algorithm is y = U \ (L \ b); Zh = U \ (L \ Z); 37...
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 Fall '11
 Dr.Robin
 Ring, Identity matrix, Downdating

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