Kensuke Aishima

On Convergence of the DQDS Algorithm for Singular Value Computation

We prove global convergence, in exact arithmetic, for the differential quotient difference algorithm that is currently implemented in LAPACK for the computation of the singular values of a bidiagonal matrix. Our results cover any shift strategy that preserves positivity. We also show that the asymptotic rate for the Johnson shift is 3/2.

dqds with Aggressive Early Deflation

The dqds algorithm computes all the singular values of an

Superquadratic convergence of DLASQ for computing matrix singular values

n \times n

A shift strategy for superquadratic convergence in the dqds algorithm for singular values

bidiagonal matrix to high relative accuracy in

Rigorous proof of cubic convergence for the dqds algorithm for singular values

O(n^2)

A Wilkinson-like multishift QR algorithm for symmetric eigenvalue problems and its global convergence

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.

Global convergence of the restarted Lanczos and Jacobi---Davidson methods for symmetric eigenvalue problems

DLASQ is a routine in LAPACK for computing the singular values of a real upper bidiagonal matrix with high accuracy. The basic algorithm, the so-called dqds algorithm, was first presented by Fernando-Parlett, and implemented as the DLASQ routine by Parlett-Marques. DLASQ is now recognized as one of the most efficient routines for computing singular values. In this paper, we prove the asymptotic superquadratic convergence of DLASQ in exact arithmetic.

Orthogonal polynomial approach to estimation of poles of rational functions from data on open curves

A new shift strategy is proposed for the differential quotient difference with shifts (dqds) algorithm for the computation of singular values of bidiagonal matrices. While maintaining global convergence, the proposed shift realizes asymptotic superquadratic convergence of the dqds algorithm.

On convergence of iterative projection methods for symmetric eigenvalue problems

Fernando and Parlett observed that the dqds algorithm for singular values can be made extremely efficient with Rutishauser’s choice of shift; in particular it enjoys “local” (or one-step) cubic convergence at the final stage of iteration, where a certain condition is to be satisfied. Their analysis is, however, rather heuristic and what has been shown is not sufficient to ensure asymptotic cubic convergence in the strict sense of the word. The objective of this paper is to specify a concrete procedure for the shift strategy and to prove with mathematical rigor that the algorithm with this shift strategy always reaches the “final stage” and enjoys asymptotic cubic convergence.

A Survey on Convergence Theorems of the dqds Algorithm for Computing Singular Values

In 1989, Bai and Demmel proposed the multishift QR algorithm for eigenvalue problems. Although the global convergence property of the algorithm (i.e., the convergence from any initial matrix) still remains an open question for general nonsymmetric matrices, in 1992 Jiang focused on symmetric tridiagonal case and gave a global convergence proof for the generalized Rayleigh quotient shifts. In this paper, we propose Wilkinson-like shifts, which reduce to the standard Wilkinson shift in the single shift case, and show a global convergence theorem.