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