ON HYPERBOLIC TRIANGULARIZATION: STABILITY AND
PIVOTING
*
MICHAEL STEWART
†
AND
G. W. STEWART
‡
SIAM J. M
ATRIX
A
NAL.
A
PPL
.
c
±
1998 Society for Industrial and Applied Mathematics
Vol. 19, No. 4, pp. 847–860, October 1998
001
Abstract.
This paper treats the problem of triangularizing a matrix by hyperbolic Householder
transformations. The stability of this method, which ﬁnds application in block updating and fast
algorithms for Toeplitzlike matrices, has been analyzed only in special cases. Here we give a gen
eral analysis which shows that two distinct implementations of the individual transformations are
relationally stable. The analysis also shows that pivoting is required for the entire triangularization
algorithm to be stable.
Key words.
hyperbolic transformation, triangularization, relational stability, pivoting
AMS subject classiﬁcations.
15A23, 65F, 65G05
PII.
S0895479897319581
1. Introduction.
Let
A
be a positive deﬁnite matrix of order
p
and let
R
T
R
be
its Cholesky factorization. Given an
m
×
p
matrix
X
, the
Cholesky updating problem
is to compute the Cholesky factorization
ˆ
R
T
ˆ
R
=
ˆ
A
≡
A
+
X
T
X
from that of
A
. Since
X
T
X
is positive semideﬁnite,
ˆ
A
is positive deﬁnite and always
has a Cholesky factor.
It is well known that the Cholesky updating problem can be solved by orthogonal
triangularization. Speciﬁcally, there is an orthogonal matrix
Q
such that
Q
T
±
R
X
²
=
±
ˆ
R
0
²
,
where
ˆ
R
is upper triangular. From the orthogonality of
Q
, it follows that
ˆ
R
T
ˆ
R
=
±
R
X
²
T
QQ
T
±
R
X
²
=
A
+
X
T
X,
so that
ˆ
R
T
ˆ
R
is the required Cholesky factorization. The matrix
Q
is usually generated
as a product of Householder transformations or plane rotations. For details see, e.g.,
[11].
Now let
Y
be an
n
×
p
matrix. The
Cholesky downdating problem
is to calculate
the Cholesky factor
ˆ
R
of
ˆ
A
=
A

Y
T
Y
from that of
A
. The downdating problem is
known to be diﬃcult. An obvious problem is that
ˆ
A
can be indeﬁnite, in which case
the problem has no (real) solution. A more subtle problem is that information present
*
Received by the editors April 4, 1997; accepted for publication (in revised form) November 26,
1997; published electronically May 7, 1998.
http://www.siam.org/journals/simax/194/31958.html
†
Computer Sciences Laboratory, Research School of Information Sciences and Engineering, Aus
tralian National University, Canberra ACT 0200, Australia. The work of this author was supported
in part by the Department of the Air Force under grant F4962209510525P00001.
‡
Department of Computer Science and Institute for Advanced Computer Studies, University of
Maryland, College Park, MD 20742 ([email protected]). The work of this author was supported
by National Science Foundation grant CCR9503126.
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 Numerical Analysis, Matrices, Orthogonal matrix, Matrix decomposition, hyperbolic householder

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