Interval Householder Method for Complex Linear
BAKYT S. DJANYBEKOV
Institute of Computational Technologies of SB RAS, 6, Lavrentiev avenue, 630090 Novosibirsk,
Russia, e-mail: email@example.com
(Received: 30 October 2004; accepted: 3 March 2005)
Abstract. We consider interval Householder method for outer estimation of solution sets for interval
linear algebraic systems with complex interval parameters. A numerical example is presented showing
that interval Householder method may work better than interval Gaussian algorithm.
We consider an interval linear algebraic system (ILAS)
with complex interval n
and complex interval n-vector
right-hand side. Its solution set is deﬁned in traditional way:
)(Ax = b)}
Even for real interval linear systems, the solution set of ILAS does not have a
simple description in the general case. Therefore, it makes sense to conﬁne ourselves
to determining some estimates of the original solution set. We will consider its outer
interval estimates or enclosures,taken in the form of interval vectors (boxes) that
entirely contain the solution set.
Interval Householder method is one of the ways for computing outer estimates
of the solution sets to interval linear systems . Its real version is also known as
reﬂection method (in particular, in Russian literature, see , ). The essence of
the point Householder method is to reduce the matrix of the system to a triangu-
lar form by a sequence of the orthogonal reﬂection transformations, and interval
Householder method is the natural interval extension of the original point algo-
rithm. Compared with interval Gauss method for the solution of interval linear
systems , interval Householder method produces narrower outer estimates of
the solution sets in case the interval matrix is “not too wide.” Moreover, interval
Householder method allows us to solve the problems that interval Gauss method
cannot be applied for.
Thsi paper was presented at IMCP’04 workshop (see Reliable Computing 11 (5)(2005)).