dqds with Aggressive Early Deflation

Abstract

Abstract. 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.Key words. aggressive early deflation, dqds, singular values, bidiagonal matrix AMS subject classifications. 65F15, 15A181. Introduction. The differential quotient difference with shifts (dqds) algorithm computes all the singular values of an n-by-n bidiagonal matrix to high relative accuracy in O(n 2 ) cost [11]. Its efficient implementation has been developed and is now available as a LAPACK subroutine DLASQ [30]. Because of its guaranteed relative accuracy and efficiency, dqds has now replaced the QR algorithm [7], which had been the default algorithm to compute the singular values of a bidiagonal matrix. The standard way of computing the singular values of a general matrix is to first apply suitable orthogonal transformations to reduce the matrix to bidiagonal form, then use dqds [6]. dqds is also a major computational kernel in the MRRR algorithm for computing orthogonal eigenvectors of a symmetric tridiagonal matrix [8,9,10] and the singular value decomposition of a bidiagonal matrix [35] in O(n 2 ) cost. The aggressive early deflation strategy, introduced in [5], is known to greatly improve the performance of the Hessenberg QR algorithm for computing the eigenvalues of a general square matrix by deflating converged eigenvalues long before a conventional deflation strategy does. The primary contribution of this paper is the proposal of two deflation strategies for dqds based on aggressive early deflation. The first strategy is a direct specialization of aggressive early deflation to dqds. The second strategy, which takes full advantage of the bidiagonal structure of the matrix, is computationally more efficient. We present a detailed mixed forward-backward stability analysis that proves the second strategy guarantees high relative accuracy of all the computed singular values. The results of extensive numerical experiments demonstrate that performing aggressive early deflation significantly reduces the solution time of dqds in many cases. We observed speedups of up to a factor 50, and in all our experiments the second strategy was at least as fast as DLASQ for any matrix larger than 3000.

Restructuring the Tridiagonal and Bidiagonal QR Algorithms for Performance

“…Alternatively, in [Dhillon 1997] it is suggested that, if eigenvalues are computed first, a "perfect shift" can be incorporated in the QR algorithm that causes deflation after only one iteration, though the same document uncovers certain challenges associated with this approach. -"Aggressive early deflation" has been developed to accelerate the QR algorithm for upper Hessenberg matrices [Braman et al 2001b], and the authors of [Nakatsukasa et al 2012] have extended the approach to the dqds algorithm. Thus, it may be possible to incorporate this technique into our restructured QR algorithm.…”

An Improved DQDS Algorithm

“…Option B. The procedure invoked in the aggressive early deflation algorithm, see [12], is useful here. In our implementation we use this option since we found it usually saves some floating point operations over Option A.…”

“…To be more specific, let ψ = √ x. The bulge moves upward by one position by applying a Givens transformation from the right [12]:…”

“…One critical technique responsible for its success is the rather complex shift strategy. An aggressive early deflation (AED) strategy is suggested in [12] to further enhance performance for large matrices.…”

“…By combining V5 with AED, a more efficient algorithm is obtained in section 6. This hybrid algorithm would be denoted by HDLASQ and is shown to be faster than DLASQ, V5 and AggDef2 (proposed in [12]) for both the easy and difficult matrices. This paper is organized as follows.…”