 
Summary:   
SIAM J. NUMER. ANAL.
Vol. 20. No. 1.Februar)' 1983
.e> 1983 Soc:icl)' for Indultrial and App\ied Mathemalic.
003l14:!9/83':!OOI0014 SOI.:!5./(1
A QUADRATlCALL Y CONVERGENT
KRA\\'CZYKLIKE ALGORITHM*
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. 187201]. 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.
(0)
1. Introduction. In 1969 R. Krawczyk[2] 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
