A Multidimensional Interval Newton Method
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(Received: 16 May 2005; accepted: 20 October 2005)
Abstract. We consider various details in multidimensional interval Newton methods and give an
algorithm based on consideration of them. The details include choice of points of expansion, com-
puting and reusing Jacobians, and choice of preconditioner. The resulting algorithm differs in several
ways from standard procedures.
Assume we wish to solve a system of nonlinear equations using an interval Newton
method. An iterated step involves linearization of the equations using a Taylor
expansion about a point in a given box, see Section 2. For simplicity, we assume
that the functions deﬁning the system have continuous derivatives of whatever order
In current practice, the point of expansion is generally chosen to be the center of
the current box or a point in the box where the function to be zeroed is expected to
be small. One purpose of this paper is to show that use of other points can provide
greater efﬁciency. We consider using more than one point of expansion for a given
Newton step, see Section 11.
In order to solve the linearized equations, it is common practice to precondition
the system using an approximate inverse of the center of the interval Jacobian J.We
consider using an inverse of a different matrix contained in J, see Section 12.
When little or no progress is made by an interval Newton method, the current
box is split into subboxes. This can cause generation of a large number of subboxes
of the initial box. As a result, a large amount of computing can be necessary.
A primary goal of the procedure we give is to reduce the number of splittings.
Automatic differentiation programs are generally available; but automatic gen-
eration of slope expansions is not. Therefore, we assume that linearization of
nonlinear functions is done using derivatives. If slopes are used, our procedure can
be modiﬁed appropriately.
We give an interval Newton algorithm in Section 14 and discuss its parts in
Sections 15 and 16. Numerical results are given in Section 17.
Reliable Computing (2006) 12: 253–272