STABILITY ISSUES IN THE FACTORIZATION OF STRUCTURED
MATRICES
*
MICHAEL STEWART
†
AND
PAUL VAN DOOREN
‡
SIAM J. M
ATRIX
A
NAL.
A
PPL.
c
±
1997 Society for Industrial and Applied Mathematics
Vol. 18, No. 1, pp. 104–118, January 1997
009
Abstract.
This paper provides an error analysis of the generalized Schur algorithm of Kailath
and Chun [
SIAM J. Matrix Anal. Appl.
, 15 (1994), pp. 114–128]—a class of algorithms which can
be used to factorize Toeplitzlike matrices, including blockToeplitz matrices, and matrices of the
form
T
T
T
, where
T
is Toeplitz. The conclusion drawn is that if this algorithm is implemented with
hyperbolic transformations in the factored form which is well known to provide numerical stability in
the context of Cholesky downdating, then the generalized Schur algorithm will be stable. If a more
direct implementation of the hyperbolic transformations is used, then it will be unstable. In this
respect, the algorithm is analogous to Cholesky downdating; the details of implementation of the
hyperbolic transformations are essential for stability. An example which illustrates this instability is
given. This result is in contrast to the ordinary Schur algorithm for which an analysis by Bojanczyk,
Brent, De Hoog, and Sweet [
SIAM J. Matrix Anal. Appl.
, 16 (1995), pp. 40–57] shows that the sta
bility of the algorithm is not dependent on the implementation of the hyperbolic transformations.
Key words.
Schur algorithm, structured matrices, Toeplitz matrices, stability
AMS subject classiﬁcations.
65F05, 65F30, 15A06, 15A23
PII.
S089547989528692X
1. Introduction.
The Schur algorithm is a popular and fast method for the
Cholesky factorization of a square, positivedeﬁnite Toeplitz matrix
T
. It performs
reliably, and in [3] it was shown to be stable in the sense that if the algorithm runs
to completion and
ˆ
C
is the computed Cholesky factor,
k
T

ˆ
C
T
ˆ
C
k
is guaranteed
to be small. This paper will perform a similar stability analysis which applies to
several special cases of the generalized Schur algorithm [10]. In its full generality,
the generalized Schur algorithm can be adapted to the factorization of a wide variety
of structured matrices. The analysis given here is primarily of interest for block
Toeplitz and Toeplitzblock matrices, as well as for matrices of the form
T
T
T
, where
T
is rectangular and Toeplitz. The key notion behind the general algorithm is the
concept of displacement rank [10].
One of the most signiﬁcant examples is the Cholesky factorization of
T
T
T
. This
factor is also the factor
R
in the
QR
factorization of the rectangular Toeplitz matrix
T
, and the obvious application of this fact to the solution of Toeplitz least squares
problems is explored in [1]. However, the analysis given there assumes the use of the
algorithm presented in [2] rather than the generalized Schur algorithm. The basic
idea is to obtain
R
without bothering about ﬁnding
Q
, thus avoiding any problems
associated with the loss of orthogonality which are common to all fast Toeplitz
QR
algorithms. The method of seminormal equations, possibly with iterative reﬁnement,