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SIAM J. NUMER. ANAL.
Vol. 20. No. 1.Februar)' 1983
.e> 1983 Soc:icl)' for Indultrial and App\ied Mathemalic.
A QUADRATlCALL Y CONVERGENT
G. ALEFELDt AND L. PLATZÖDER;
Abitract. In this paper we introducc a method for computing a solution of a nonlinear system. which
is similar to that proposcd by R. Krawczyk [Computing. 4 (1969). pp. 187-201]. Dur method. however.
necds coniiderably less work per step. Starting with an interval vector. we give a criterion under which
the method is convergent to the soh~tionof the system if a solution is contained in the interval vector. lf
tbc starting vcctor contains no solution then the method will break down after a finite number of steps.
1. Introduction. In 1969 R. Krawczyk introduced the operator
k(x, Y)= m(x)- y!(m (x» + (J::- YI'(x»(x- m (x»,
where I: ~c Vn(R)~ Vn(R) denotes a mapping I, xs;\8 is an interval vector, m(x) is
the center of x, Yis areal n x n matrix, J::denotes the unit matrix and ('(x) is the
interval arithmetic evaluation of the derivative of ( over the interval vector x. The
result k(x, Y) is an interval vector. Using this operator, Krawczyk considered the