Reliable Computing 8: 229–237, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
An Approach to Overcome Division by Zero in
the Interval Gauss Algorithm
Dedicated to Prof. G
otz Alefeld on occasion of his 60th birthday
ur Angewandte Mathematik, Universit
at Karlsruhe, Karlsruhe, Germany,
(Received: 22 June 2001; accepted: 21 March 2002)
Abstract. We will discuss some modiﬁcations of the interval Gauss algorithm, so that it becomes
feasible in a more general setting, particularly for totally non-negative matrices.
The Interval Gauss Algorithm (IGA) is an option for solving a system of linear
interval equations. As is well-known, (IGA) can break down because the interval for
a pivot element contains 0, even though the usual Gauss Algorithm works for every
point matrix contained in the coefﬁcient interval matrix (see subsequent examples).
In this paper, we will show that if sufﬁciently accurate interval approximations of
the principal minors of the coefﬁcient matrix are known, then this problem can be
overcome (Theorem 3.1). Generally, the problem of approximating determinants of
an interval matrix is at least as difﬁcult as the original problem of solving a system of
linear interval equations, so nothing is gained. For some interval matrices however,
most importantly for totally non-negative interval matrices, the determinant can be
calculated exactly (Theorem 5.1). Consequently, this approach provides a method to
solve a system of interval linear equations having a totally non-negative coefﬁcient
matrix by using a modiﬁed Interval Gauss Algorithm for those cases, where (IGA)
breaks down. This procedure can however only be practical, if a given matrix can
be checked for total non-negativity easily. To this end, we cite both a simple test
providing sufﬁcient conditions and a more difﬁcult test providing both necessary
and sufﬁcient conditions for a matrix to be totally non-negative (Theorem 6.1). This
yields a practical procedure: check if the interval matrix is totally non-negative by
one of the tests, perform (IGA), and if one of the pivots contains 0, use Theorem 5.1
to obtain a better estimate of the pivot not containing 0 and continue.
Totally non-negative matrices occur in statistics, approximation theory, geomet-
ric modelling, the theory of splines and other areas. For speciﬁc applications check
for example the contributions to a conference on total positivity, , and the ref-