Geodesic
The $\mathrm{MR}^{3}$-GK algorithm for the bidiagonal SVD
Electronic transactions on numerical analysis, Tome 39 (2012), pp. 1-21.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 65F30, 65F15, 65G50, 15A18
Keywords: bidiagonal matrix, singular value decomposition, MRRR algorithm, theory and implementation, golub, kahan matrix 
@article{ETNA_2012__39__a25,
     author = {Willems, Paul R. and Lang, Bruno},
     title = {The $\mathrm{MR}^{3}${-GK} algorithm for the bidiagonal {SVD}},
     journal = {Electronic transactions on numerical analysis},
     pages = {1--21},
     publisher = {mathdoc},
     volume = {39},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a25/}
}
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Willems, Paul R.; Lang, Bruno. The $\mathrm{MR}^{3}$-GK algorithm for the bidiagonal SVD. Electronic transactions on numerical analysis, Tome 39 (2012), pp. 1-21. http://geodesic.mathdoc.fr/item/ETNA_2012__39__a25/