SIAM J, NUMER, ANAL.
Vol. 21. 1'10.2, April 1984
@ 1984 Socie:y for Il1dustria! and Applied Mathematics
ON THE CONVERGENCE OF SOME INTERVAL-ARITHMETIC
MODIFICATIONS OF NEWTON'S METHOD*
.Abstract. In this paper we consider the meanwhile weJl-known interval-arithmetic modifications of the
Newton's method and of the so-calJed simplifiedNewton's method for sohringsystemsof nonlinear equations.
Starting with an interval-vector which contains a zero. we give for the first time sufficientconditions for the
convergencc of these methods to this solution. If thc starting interval-vector contains no solution, then
under the very same conditions the methods under consideration will break down after a finite number of
steps. The interval-arithmetic 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 interval-arithmetic evaluation
f'(Xk) of the derivative exists, then, if 0 Ef'C'(k), the method
m(Xk) EX\ (m(Xk) ElR),
yk+! =f (X