• Corpus ID: 209404986

# Cross product-free matrix pencils for computing generalized singular values

@article{Zwaan2019CrossPM,
title={Cross product-free matrix pencils for computing generalized singular values},
author={Ian N. Zwaan},
journal={ArXiv},
year={2019},
volume={abs/1912.08518}
}
• I. Zwaan
• Published 18 December 2019
• Mathematics
• ArXiv
It is well known that the generalized (or quotient) singular values of a matrix pair $(A, C)$ can be obtained from the generalized eigenvalues of a matrix pencil consisting of two augmented matrices. The downside of this reformulation is that one of the augmented matrices requires a cross products of the form $C^*C$, which may affect the accuracy of the computed quotient singular values if $C$ has a large condition number. A similar statement holds for the restricted singular values of a matrix…
2 Citations

## Figures from this paper

Two harmonic Jacobi-Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair
• Computer Science
ArXiv
• 2022
Two harmonic extraction based Jacobi–Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair and converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values.

## References

SHOWING 1-10 OF 16 REFERENCES
Using cross-product matrices to compute the SVD
• Z. Jia
• Computer Science
Numerical Algorithms
• 2006
It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$.
The restricted singular value decomposition: properties and applications
• Mathematics
• 1991
The restricted singular value decomposition (RSVD) is the factorization of a given matrix, relative to two other given matrices. It can be interpreted as the ordinary singular value decomposition
On choices of formulations of computing the generalized singular value decomposition of a large matrix pair
• Computer Science, Mathematics
Numerical Algorithms
• 2020
The concern in this paper is, in finite precision arithmetic, which generalized eigenvalue formulation is numerically preferable to compute the desired GSVD components more accurately.
Computing the generalized singular values/vectors of large sparse or structured matrix pairs
A numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair based on the CS decomposition and the Lanczos bidiagonalization process is presented.
The restricted singular value decomposition of matrix triplets
In this paper the concept of restricted singular values of matrix triplets is introduced. A decomposition theorem concerning the general matrix triplet $( A,B,C )$, where \$A \in \mathcal{C}^{m \times
Towards a Generalized Singular Value Decomposition
• Mathematics
• 1981
We suggest a form for, and give a constructive derivation of, the generalized singular value decomposition of any two matrices having the same number of columns. We outline its desirable
Computing the Generalized Singular Value Decomposition
• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 1993
A new numerical method for computing the GSVD of two matrices A and B is presented, a variation on Paige''s method, which differs from previous algorithms in guaranteeing both backward stability and convergence.
A Golub–Kahan-Type Reduction Method for Matrix Pairs
• Mathematics
J. Sci. Comput.
• 2015
A novel method for reducing a Pair of large matrices A and B to a pair of small matrices H,K is described, an extension of Golub–Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix.