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Existence of Solution~ and Iterations for Nonlinear Equations

Summary: Existence of Solution~ and Iterations for
Nonlinear Equations
by G. Alefeld, Karlsruhe University
1. Introduction. In this paper we discuss the use of inter-
val analysis in order to prove the existence of solutions of
an equation. In Chapter 2 we repeat the generalization of
the bisection process using interval arithmetic tools. The
use of the Brouwer fixed-point theorem is demonstrated in
Chapter 3. We show in Example 1 that by using interval
arithmetic it is sometimes possible to improve known
existence statements. since the proof of the Brouwer fixed-
point theorem is nontrivial it seems worthwhile to investi-
gate if one can prove the existence of fixed points by using
interval arithmetic tools alone. Some ideas in this direc-
tion are described in Chapter 4. In the final Chapter 5 the
Interval-Newton-Method is reconsidered again and a new
statement concerning the order of convergence is given. The
terminology used in this paper is the same as in [5].
2. Bisection. It is well-known that if for areal continuous
function f : ffi ~ ffi there exist reals a and b , a < b ,


Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe


Collections: Mathematics