Reliable Computing 8: 123–130, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
A Parallelized Version of the Covering Algorithm
for Solving Parameter-Dependent Systems of
PALURI S. V. NATARAJ and AIRANI KALATHIL PRAKASH
Systems and Control Engineering Group, Department of Electrical Engineering,
IIT Bombay 400 076, India, e-mail: email@example.com
(Received: 30 October 2000; accepted: 2 April 2001)
Abstract. The so-called covering algorithm for enclosing the solution set of parameter—dependent
systems of nonlinear equations has been recently proposed by Neumaier (The Enclosure of Solutions
of Parameter Dependent Systems of Equations, in: Moore, R. E. (ed.), Reliability in Computing:
The Role of Interval Methods in Scientiﬁc Computations, Academic Press, 1988). However, in the
covering algorithm, only one box is processed in each iteration. This paper presents a parallelized
version of the covering algorithm, in which all boxes present are processed simultaneously in each
iteration. It is shown through several examples that this strategy results in speed-up of the algorithm
by several orders of magnitude, particularly so in demanding problems. The proposed parallelized
version can be run even on ordinary computers, i.e., it does not require a parallel computer.
This paper addresses the problem of ﬁnding in a given box (i.e., a rectangular
parallelipiped) all solutions of a nonlinear system with more variables than equa-
tions. This problem is clearly of a broad scope and has numerous applications in
engineering and sciences.
The problem can sometimes be solved by one of the following methods :
(i) random search, (ii) an exhaustive grid search on the given box, (iii) more
specialized or ad hoc methods, such as the Jenkins-Traub method for ﬁnding all
roots of a single polynomial, and (iv) homotopy continuation methods . The
interested reader is refereed to ,  for a discussion and comparision of these
In the frame work of interval analysis , Neumaier proposed the so-called
covering algorithm  to solve the problem. Consider a ﬁnite-dimensional system
of nonlinear equations of the form