 
Summary: International Series of Numerical Mathematics, Vol. 96, @ 1991 Birkhäuser Verlag Basel 9
AN ENCLOSURE METHOD WITH HIGHER ORDER OF CONVERGENCE
APPLICA TIONS TO THE ALGEBRAIC EIGENV ALUE PROBLEM
G. Alefeld , University of Karlsruhe
B. Illg, University of Karlsruhe
F. Potra, University of Iowa
Dedicated to L. Collatz on the occassion 0f his 80th birthday
1. Introduction
In [1] we have considered the nonlinear equation f(x) = 0 where f is a continuous
dilierentiable real function of a real variable. We suppose that f is strictly monotone
on an interval XO. Without loss of generality we may assume that f is strictly
increasing on XO. We assume that by using interval arithmetic methods it is possible
to compute two positive numbers tl' t2 such that 0 < tl ~ f' (x) ~ t2 for all x E XO.
Let us denote by L the interval [tl,t2]' We suppose that the derivative f' (x) EIR,
x E XO, has an interval extension f' (X) , X S XO, satisfying the following conditions
10 G. AIefeld et al.
f' (x) E f I (X) ,
f'(X) ~ f'(Y),
d(f' (X)) ~ c d(X) ,
XEX~X<>
