Reliable Computing 8: 249–265, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Sparse Systems in Fixed Point Form
RAMON E. MOORE
40 Orchard Dr., Worthington, OH 43085, USA, e-mail: firstname.lastname@example.org
(Received: 10 December 2001; accepted: 12 April 2002)
Abstract. 1) A method is shown for reducing dimension of sparse systems of equations given in
ﬁxed point form. Economic models provide examples. 2) A given box to be searched for solutions
can be reduced using interval intersections. 3) Interval Newton-like methods using either derivatives
or slopes can be applied to the reduced system.
1. Reduction of Dimension
There are methods for reducing dimension for sparse systems of equations given in
ﬁxed point form. Such sparse systems occur naturally in economic models such as
the three to be discussed here as examples:
1) The largest of more than 200 “irreducible” (coupled) blocks in a model of the
national economy of India has dimension 30, but will be reduced to an equivalent
problem of dimension 3—that is three equations in three unknowns.
2) An irreducible block of dimension 83 in a model of world production and
consumption of sugar will be reduced to one equation in a single variable.
3) A block of dimension 99 from an economic model called FAIR (see ),
obtained from MIT, will be reduced to a system of dimension 6.
Such reductions in the number of variables in a system to be solved can improve
efﬁciency of computational methods. Efﬁciency can be important no matter how
fast machines are, because it frees up time on the machines to do other work as
well, and hence be more productive.
We restrict the class of systems we are considering here to those which can be
put into ﬁxed point form, x = F(x). This form is typical for the systems of equations
for the “endogenous” variables in an economic model, the other variables, the
“exogenous” ones have given “input” values, and we wish to solve for the values
of the endogenous variables.
In dynamic (time dependent) economic models, current values of endogenous
variables are the unknowns in the equations of the system, and the coefﬁcients may
depend on previous values of the endogenous variables as well as on values of
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