# Computing the Generalized Singular Value Decomposition

@article{Bai1993ComputingTG, title={Computing the Generalized Singular Value Decomposition}, author={Zhaojun Bai and James Demmel}, journal={SIAM J. Sci. Comput.}, year={1993}, volume={14}, pages={1464-1486} }

We present a new numerical method for computing the GSVD [36, 27] of two matrices A and B. This method is a variation on Paige''s method [30]. It differs from previous algorithms in guaranteeing both backward stability and con- vergence. There are two innovations. The first is a new pre- processing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 by 2 triangular GSVD algorithm, which constitutes the inner loop of Paige''s method. We…

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## References

SHOWING 1-10 OF 45 REFERENCES

On Jacobi methods for singular value decompositions

- Mathematics
- 1987

An improvement of the Jacobi singular value decomposition algorithm is proposed. The matrix is first reduced to a triangular form. It is shown that the row-cyclic strategy preserves the…

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 CS and the generalized singular value decompositions

- Computer Science
- 1985

Stewart has given an algorithm that uses the LINPACK SVD algorithm together with a Jacobitype "clean-up" operation on a cross-product matrix, which is equally stable and fast but avoids the cross product matrix.

A Numerical Algorithm for Computing the Restricted Singular Value Decomposition of Matrix Triplets.

- Mathematics
- 1992

Computing the generalized singular value decomposition

- Computer Science
- 1986

With the correct choice of ordering the algorithm can be implemented using systolic array processors (Gentleman, personal communication), and can also be used to compute any CS decomposition of a unitary matrix.

The cyclic Jacobi method for computing the principal values of a complex matrix

- Mathematics
- 1960

is diagonal (T denotes the transpose), then the main diagonal of A is made up of the numbers Xi in some order. If it is desired to compute the Xi numerically, this result is of no immediate use,…

Computing the singular value decompostion of a product of two matrices

- Mathematics
- 1986

An algorithm is developed for computing the singular value decomposition of a product of two general matrices without explicitly forming the product. The algorithm is based on an earlier Jacobi-like…

Numerical treatment of restricted gauss-markov model 1

- Mathematics
- 1988

The singular value decomposition (SVD) has been widely used in the ordinary linear model and other statistical problems. In this paper, we shall introduce the generalized singular value decomposition…

Perturbation Analysis for the Generalized Singular Value Problem

- Mathematics
- 1983

This paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of…