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Modifiable low‐rank approximation to a matrix
Author(s) -
Barlow Jesse L.,
Erbay Hasan
Publication year - 2009
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.651
Subject(s) - mathematics , rank (graph theory) , triangular matrix , matrix (chemical analysis) , singular value decomposition , combinatorics , degeneracy (biology) , decomposition , matrix decomposition , low rank approximation , orthogonal matrix , algorithm , algebra over a field , pure mathematics , eigenvalues and eigenvectors , physics , orthogonal basis , ecology , bioinformatics , materials science , composite material , quantum mechanics , tensor (intrinsic definition) , invertible matrix , biology
A truncated ULV decomposition (TULVD) of an m × n matrix X of rank k is a decomposition of the form X = ULV T + E , where U and V are left orthogonal matrices, L is a k × k non‐singular lower triangular matrix, and E is an error matrix. Only U , V , L , and ∥ E ∥ F are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in ( SIAM J. Matrix Anal. Appl. 2005; 27 (1):198–211) that reduces ∥ E ∥ F , detects rank degeneracy, corrects it, and sharpens the approximation. Copyright © 2009 John Wiley & Sons, Ltd.