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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 . orming A 1 from the LU decomposition takes O ( n 3 ) operations, but forming A 1 b as U \ ( L \ b ) uses forward- and back-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 ). (orming 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 algo-rithm is y = U \ (L \ b); Zh = U \ (L \ Z); 37...
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- Fall '11