### Computing the Generalized Singular Value Decomposition

**Authors:**

Bai, Zhaojun

Demmel, James W.
**Technical Report Identifier:** CSD-92-720
**December 1992**

CSD-92-720.pdf

CSD-92-720.ps

**Abstract:** We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices *A* and *B*. There are two innovations. The first is a new preprocessing 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 algorithm. We present proofs of stability and high accuracy of the 2 by 2 GSVD algorithm, and demonstrate it using examples on which all previous algorithms fail.

Keywords: generalized singular value decomposition, CS decomposition, matrix decomposition, Jacobi algorithm, Kogbetliantz algorithm.