The QR algorithm for unitary Hessenberg matrices
Journal of Computational and Applied Mathematics
Accurate singular values of bidiagonal matrices
SIAM Journal on Scientific and Statistical Computing
On a block implementation of Hessenberg multishift QR iteration
International Journal of High Speed Computing
Deferred shifting schemes for parallel QR methods
SIAM Journal on Matrix Analysis and Applications
Forward instability of tridiagonal QR
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
A new parallel chasing algorithm for transforming arrowhead matrices to tridiagonal form
Mathematics of Computation
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
Quadratic Residual Bounds for the Hermitian Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
The Multishift QR Algorithm. Part II: Aggressive Early Deflation
SIAM Journal on Matrix Analysis and Applications
Orthogonal Eigenvectors and Relative Gaps
SIAM Journal on Matrix Analysis and Applications
Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers
ACM Transactions on Mathematical Software (TOMS)
On Convergence of the DQDS Algorithm for Singular Value Computation
SIAM Journal on Matrix Analysis and Applications
The Effect of Aggressive Early Deflation on the Convergence of the QR Algorithm
SIAM Journal on Matrix Analysis and Applications
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
Superquadratic convergence of DLASQ for computing matrix singular values
Journal of Computational and Applied Mathematics
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The dqds algorithm computes all the singular values of an $n \times n$ 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.