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• W2093972952 abstract "Bidiagonal reduction is the preliminary stage for the fastest stable algorithms for computing the singular value decomposition (SVD) now available. However, the best-known error bounds on bidiagonal reduction methods on any matrix are of the form [ A + delta A = UBV^T , ] [ |delta A|_2 leq varepsilon_M f(m,n) |A|_2, ] where B is bidiagonal, U and V are orthogonal, $varepsilon_M$ is machine precision, and f(m,n) is a modestly growing function of the dimensions of A. A preprocessing technique analyzed by Higham [Linear Algebra Appl., 309 (2000), pp. 153--174] uses orthogonal factorization with column pivoting to obtain the factorization [ A=Q left( begin{array}{c} C^T 0 end{array} right) P^T, ] where Q is orthogonal, C is lower triangular, and P is permutation matrix. Bidiagonal reduction is applied to the resulting matrix C. To do that reduction, a new Givens-based bidiagonalization algorithm is proposed that produces a bidiagonal matrix B that satisfies $C + delta C = U (B + delta B ) V^T$ where $delta B$ is bounded componentwise and $delta C$ satisfies a columnwise bound (based upon the growth of the lower right corner of C) with U and V orthogonal to nearly working precision. Once we have that reduction, there is a good menu of algorithms that obtain the singular values of the bidiagonal matrix B to relative accuracy, thus obtaining an SVD of C that can be much more accurate than that obtained from standard bidiagonal reduction procedures. The additional operations required over the standard bidiagonal reduction algorithm of Golub and Kahan [J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205--224] are those for using Givens rotations instead of Householder transformations to compute the matrix V, and 2n3/3 flops to compute column norms." @default.
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• W2093972952 creator A5033542875 @default.
• W2093972952 date "2002-01-01" @default.
• W2093972952 modified "2023-09-26" @default.
• W2093972952 title "More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition" @default.
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• W2093972952 doi "https://doi.org/10.1137/s0895479898343541" @default.
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