# Computing the Generalized Singular Value Decomposition

### Zhaojun Bai and James W. Demmel

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-91-645

August 1991

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD-91-645.pdf

We present a new numerical method for computing the GSVD of two matrices
*A* and
*B*. This method is a variation on Paige's method. It differs from previous algorithms in guaranteeing both backward stability and convergence. 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 method. We present proofs of stability and convergence of our method, and demonstrate examples on which all previous algorithms fail.

BibTeX citation:

@techreport{Bai:CSD-91-645, Author = {Bai, Zhaojun and Demmel, James W.}, Title = {Computing the Generalized Singular Value Decomposition}, Institution = {EECS Department, University of California, Berkeley}, Year = {1991}, Month = {Aug}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1991/6389.html}, Number = {UCB/CSD-91-645}, Abstract = {We present a new numerical method for computing the GSVD of two matrices <i>A</i> and <i>B</i>. This method is a variation on Paige's method. It differs from previous algorithms in guaranteeing both backward stability and convergence. 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 method. We present proofs of stability and convergence of our method, and demonstrate examples on which all previous algorithms fail.} }

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 1991 %@ UCB/CSD-91-645 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1991/6389.html %F Bai:CSD-91-645