# Computing the Generalized Singular Value Decomposition

### Zhaojun Bai and James W. Demmel

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-92-720

December 1992

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/CSD-92-720.pdf

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

BibTeX citation:

@techreport{Bai:CSD-92-720, Author = {Bai, Zhaojun and Demmel, James W.}, Title = {Computing the Generalized Singular Value Decomposition}, Institution = {EECS Department, University of California, Berkeley}, Year = {1992}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6016.html}, Number = {UCB/CSD-92-720}, Abstract = {We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices <i>A</i> and <i>B</i>. There are two innovations. The first is a new preprocessing step which reduces <i>A</i> and <i>B</i> 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. <p>Keywords: generalized singular value decomposition, CS decomposition, matrix decomposition, Jacobi algorithm, Kogbetliantz algorithm} }

EndNote citation:

%0 Report %A Bai, Zhaojun %A Demmel, James W. %T Computing the Generalized Singular Value Decomposition %I EECS Department, University of California, Berkeley %D 1992 %@ UCB/CSD-92-720 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6016.html %F Bai:CSD-92-720