dqds with aggressive early deflation

Nakatsukasa, Yuji and Aishima, Kensuke and Yamazaki, Ichitaro (2011) dqds with aggressive early deflation. [MIMS Preprint]

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The dqds algorithm computes all the singular values of an $n$-by-$n$ bidiagonal matrix to high relative accuracy in $O(n^2)$ cost. Its efficient implementation is now available as a LAPACK subroutine and is the preferred algorithm for this purpose. In this paper we incorporate into dqds a technique called aggressive early deflation, which has been applied successfully to the Hessenberg QR algorithm. Extensive numerical experiments show that aggressive early deflation often reduces the dqds runtime significantly. In addition, our theoretical analysis suggests that with aggressive early deflation, the performance of dqds is largely independent of the shift strategy. We confirm through experiments that the zero-shift version is often as fast as the shifted version. We give a detailed error analysis to prove that with our proposed deflation strategy, dqds computes all the singular values to high relative accuracy.

Item Type: MIMS Preprint
Uncontrolled Keywords: aggressive early deflation, dqds, singular values, bidiagonal matrix
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Yuji Nakatsukasa
Date Deposited: 19 Dec 2011
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1688

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