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}
}
  • Z. Bai, J. Demmel
  • Published 1 November 1993
  • Computer Science, Mathematics
  • SIAM J. Sci. Comput.
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… 
Towards a more robust algorithm for computing the restricted singular value decomposition
TLDR
A new algorithm to compute the restricted singular value decomposition of dense matrices is presented, like Zha's method, but with four major innovations, including a useful quasi-upper triangular generalized Schur form that just requires orthonormal transformations to compute.
Randomized Generalized Singular Value Decomposition
TLDR
This paper uses random projections to capture the most of the action of the matrices and proposes randomized algorithms for computing a low-rank approximation of the generalized singular value decomposition of two matrices.
Rescaling the GSVD with application to ill-posed problems
TLDR
A scaling that seeks to minimize the condition number of XT, a generalized singular value decomposition of a pair of matrices, that is well suited for the solution of linear discrete ill-posed problems.
Condition Number and Backward Error for the Generalized Singular Value Decomposition
TLDR
A normwise backward error of {A, B} with respect to an approximate GSV and an associated approximate generalized singular vector group is defined, and a computable formula of the backward error is obtained.
Generalizations of the singular value and QR decompositions
  • B. Moor
  • Computer Science, Mathematics
    Signal Process.
  • 1991
The calculation of linear least squares problems
TLDR
Some recent estimates of the optimal backward error for an alleged solution to an LS problem are presented and some generalized QR factorizations which can be used to solve different generalized least squares problems are presented.
Simplified GSVD computations for the solution of linear discrete ill-posed problems
On the error analysis and implementation of some eigenvalue decomposition and singular value decomposition algorithms
TLDR
This thesis presents several new algorithms and improvements on old algorithms, analyzing them with respect to their speed, accuracy, and storage requirements, and discusses the variations on the bisection algorithm for finding eigenvalues of symmetric tridiagonal matrices.
...
...

References

SHOWING 1-10 OF 45 REFERENCES
On Jacobi methods for singular value decompositions
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
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
TLDR
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.
Computing the generalized singular value decomposition
TLDR
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
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
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
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
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
The general linear model and the generalized singular value decomposition
...
...