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Published in:
2010 | OriginalPaper | Chapter
Updating and Downdating Matrix Decompositions
Authors : Lars Eldén, Misha E. Kilmer, Dianne P. O’Leary
Published in: G.W. Stewart
Publisher: Birkhäuser Boston
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The Sherman–Morrison–Woodbury formula ([GWS-B7], p. 328) is a recipe for constructing the inverse of a matrix after it has been modified by a low-rank correction. For a matrix of size
n
×
n
that has been so modified, it enables the inverse of this matrix to be updated in time proportional to
kn
, where
k
is the rank of the correction, rather than the
n
3
time usually necessary to compute the inverse directly. This important fact has enabled a variety of algorithms, from early implementations of the simplex method for linear optimization [29] to algorithms for solving least squares problems when new data arrive.