Generalized theory and some specializations of the Region Contraction Algorithm I — Ball operation

Abstract

We describe a new algorithm named Region Contraction Algorithm for solving certain nonlinear equations, and establish the convergence of the algorithm and give an error estimation. It is shown that this general theory includes all of present existing ball iterations as special cases.

To find a zero of a quasi-strongly monotone mapping, which arises often from the field of differential equations, variational calculus and optimization etc., the authors [2] recently proposed a new algorithm called Region Contraction Algorithm (abbreviated RCA henceforth) in real Hilbert spaces. Stemming from T.E. Williamson's geometric estimation for fixed points of contractive mappings [3], the algorithm establishes a convergent iterative process which keeps well defined and automatically covers the errors by constructing a sequence of closed balls containing the zero set. Later on, proceeding in a completely different view from the authors, Wu Yujiang and Wang Deren [4] rewrited our algorithm in the language of interval analysis, and also suggested a new globally convergent scheme in the case that F is strongly monotone. It showed the authors that the RCA is almost Nickel's Ball Newton Method [1] (abbreviated BNM henceforth) except for the difference of the class of mappings to which it applies.

In this paper we develop a more general algorithm called stationary region contracting algorithm (abbreviated SRCA) with RCA, BNM and some other methods as its specializations.

In Section 1 we present the algorithm and give some basic properties in Section 2. In Section 3 we prove convergence of the algorithm and discuss some specializations in the last section.

In what follows, we always let H be a real Hilbert space with inner product (.,.), and use B(x,r) to denote the closed ball with center x and radius r.