Reliable Computing 7: 467–483, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
On the Combination of Interval Constraint
IRIN, University of Nantes, B.P. 92208, F-44322 Nantes Cedex 3, France,
(Received: 4 May 2000; accepted: 7 February 2001)
Abstract. This paper tackles the combination of interval methods for solving nonlinear systems.
A cooperative strategy of application of elementary solvers is designed in order to accelerate the
whole computation while weakening the local domain contractions. It is implemented in a prototype
solver which efﬁciently combines interval-based local consistencies and the multidimensional interval
Newton method. A set of experiments shows a gain of one order of magnitude on average with respect
1. Introduction and Related Work
Numerical constraints over continuous domains are involved in many applications
from robotics, chemistry, geometry, etc. Constraint solving algorithms are of partic-
ular importance since they are often used by optimization or differentiation engines.
Among them, branch and prune algorithms alternate domain pruning by enforcing
local consistencies to contract the variable domains, and branching to traverse the
The combination of interval-based domain pruning techniques has been dis-
cussed ever since the pioneering work of R. E. Moore (e.g., see , , ).
More recently, P. Van Hentenryck et al.  have designed the core solver of
Numerica which implements box consistency over different interval extensions of
the constraints to be processed. The essential idea from these frameworks is to com-
bine redundant computations—different reliable approximations of one quantity—
from possibly heterogeneous solvers. Though the precision of computed domains
is necessarily improved, the amount of computation time induced from the simul-
taneous use of redundant solvers must be taken care of.
In this paper, we study a particular combination of interval techniques founded
on a cooperative approach to collectively solve a constraint system. On the one
hand, hull consistency  and box consistency  have been combined in  in
order to decrease the computation time with respect to the use of box consistency
alone. On the other hand, box consistency has been combined with the multidimen-
sional interval Newton method (called thereafter interval Newton) in  where a
constraint system is linearized through ﬁrst order Taylor expansion and precondi-