Computing the generalized singular value decomposition
SIAM Journal on Scientific and Statistical Computing
A new preprocessing algorithm for the computation of the generalized singular value decomposition
SIAM Journal on Scientific Computing
Some perspectives on the eigenvalue problem
SIAM Review
Computing the generalized singular value decomposition
SIAM Journal on Scientific Computing
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Accelerating the SVD Block-Jacobi Method
Computing - Editorial: Special issue on GAMM – Workshop on Guaranteed Error-bounds for the Solution of Nonlinear Problems in Applied Mathematics
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
The Beta-Jacobi Matrix Model, the CS Decomposition, and Generalized Singular Value Problems
Foundations of Computational Mathematics
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Since its discovery in 1977, the CS decomposition (CSD) has resisted computation, even though it is a sibling of the well-understood eigenvalue and singular value decompositions. Several algorithms have been developed for the reduced 2-by-1 form of the decomposition, but none have been extended to the complete 2-by-2 form of the decomposition in Stewart's original paper. In this article, we present an algorithm for simultaneously bidiagonalizing the four blocks of a unitary matrix partitioned into a 2-by-2 block structure. This serves as the first, direct phase of a two-stage algorithm for the CSD, much as Golub-Kahan-Reinsch bidiagonalization serves as the first stage in computing the singular value decomposition. Backward stability is proved.