 
Summary: ':.,
SIAM J, NUMER, ANAL.
Vol. 21. 1'10.2, April 1984
@ 1984 Socie:y for Il1dustria! and Applied Mathematics
012
ON THE CONVERGENCE OF SOME INTERVALARITHMETIC
MODIFICATIONS OF NEWTON'S METHOD*
G. ALEFELDt
.Abstract. In this paper we consider the meanwhile weJlknown intervalarithmetic modifications of the
Newton's method and of the socalJed simplifiedNewton's method for sohringsystemsof nonlinear equations.
Starting with an intervalvector which contains a zero. we give for the first time sufficientconditions for the
convergencc of these methods to this solution. If thc starting intervalvector contains no solution, then
under the very same conditions the methods under consideration will break down after a finite number of
steps. The intervalarithmetic evaluation of the first derivative is only involvcd in these conditions.
1. Introduction. In order to motivate the resuJts of the later sections, we first
consider a single equation with one unknown. If f has a zero x* which is contained in
the interval XOs; X, where f: Xc iR~ IR, and if the intervalarithmetic evaluation
f'(Xk) of the derivative exists, then, if 0 Ef'C'(k), the method
m(Xk) EX\ (m(Xk) ElR),
yk+! =f (X
