Rankrevealing factorizations and the singular value decomposition
Authors:
Y. P. Hong and C.T. Pan
Journal:
Math. Comp. 58 (1992), 213232
MSC:
Primary 65F05
MathSciNet review:
1106970
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Abstract 
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Abstract: T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rankrevealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size with numerical rank r. The bounds derived in this paper that guarantee the existence of RRQR are all of order , in comparison with Chan's . It has been known for some time that if A is only numerically rankone deficient, then the column permutation of A that guarantees a small in the QR factorization of can be obtained by inspecting the size of the elements of the right singular vector of A corresponding to the smallest singular value of A. To some extent, our paper generalizes this wellknown result.
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 L. Foster, Rank and null space calculations using matrix decomposition without column interchanges, Linear Algebra Appl. 74 (1986), 4771. MR 822139 (87f:65052)
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 G. H. Golub, Numerical methods for solving least squares problems, Numer. Math. 7 (1965), 206216. MR 0181094 (31:5323)
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 G. H. Golub, V. Klema, and G. W. Stewart, Rank degeneracy and least squares problems, Report TR456, Dept. of Computer Science, Univ. of Maryland, 1976.
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 G. H. Golub and C. F. Van Loan, Matrix computations, 2nd ed., John Hopkins Univ. Press, 1989. MR 1002570 (90d:65055)
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 R. J. Hanson and C. L. Lawson, Solving least squares problems, PrenticeHall, 1974. MR 0366019 (51:2270)
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 R. A. Horn and C. A. Johnson, Matrix analysis, Cambridge Univ. Press, 1985. MR 832183 (87e:15001)
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 W. Kahan, Numerical linear algebra, Canad. Math. Bull. 9 (1966), 757801.
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 I. Karasalo, A criterion for truncation of the QRdecomposition algorithm for the singular linear least squares problem, BIT 14 (1974), 156166. MR 0483361 (58:3372)
 [16]
 T. Manteuffel, An interval analysis approach to rank determination in linear least squares problems, SIAM J. Sci. Statist. Comput. 2 (1981), 335348. MR 632904 (82m:65042)
 [17]
 E. Ng, A scheme for handling rank deficiency in the solution of sparse linear least squares problems, Report ORNL/TM10980, Mathematics Division, Oak Ridge National Laboratory, 1988.
 [18]
 G. W. Stewart, On the implicit deflation of nearly singular systems of linear equations, SIAM J. Sci. Statist. Comput. 2 (1981), 136140. MR 622710 (82f:65030)
 [19]
 , Rank degeneracy, SIAM J. Sci. Statist. Comput. 5 (1984), 403413. MR 740857 (85h:65086)
 [20]
 , Incremental condition calculation and column selection, Report CSTR2495, Dept. of Computer Science, Univ. of Maryland, 1990.
 [21]
 S. Van Huffel, J. Vandewalle, and A. Haegemans, An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values, J. Comput. Appl. Math. 19 (1987), 313330. MR 907802 (89c:65051)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199211069704
PII:
S 00255718(1992)11069704
Keywords:
Singular value decomposition,
rankrevealing QR factorization,
numerical rank,
numerical null space
Article copyright:
© Copyright 1992
American Mathematical Society
