Volume 39, pp. 1-21, 2012.

The ${\mbox{MR}}^{3}$-GK algorithm for the bidiagonal SVD

Paul R. Willems and Bruno Lang


Determining the singular value decomposition of a bidiagonal matrix is a frequent subtask in numerical computations. We shed new light on a long-known way to utilize the algorithm of multiple relatively robust representations, $\mbox{MR}^{\mathrm{3}}$, for this task by casting the singular value problem in terms of a suitable tridiagonal symmetric eigenproblem (via the Golub–Kahan matrix). Just running $\mbox{MR}^{\mathrm{3}}$ “as is” on the tridiagonal problem does not work, as has been observed before (e.g., by B. Großer and B. Lang [Linear Algebra Appl., 358 (2003), pp. 45–70]). In this paper we give more detailed explanations for the problems with running $\mbox{MR}^{\mathrm{3}}$ as a black box solver on the Golub–Kahan matrix. We show that, in contrast to standing opinion, $\mbox{MR}^{\mathrm{3}}$ can be run safely on the Golub–Kahan matrix, with just a minor modification. A proof including error bounds is given for this claim.

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Key words

bidiagonal matrix, singular value decomposition, MRRR algorithm, theory and implementation, Golub–Kahan matrix

AMS subject classifications

65F30, 65F15, 65G50, 15A18

Links to the cited ETNA articles

[36]Vol. 38 (2011), pp. 363-400 Paul R. Willems and Bruno Lang: Block factorizations and qd-type transformations for the $\mbox{MR}^3$ algorithm

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