Generalized Theory and Some Specializations of the Region Contraction Algorithm I - Ball Operation

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 gene­ ral theory includes all of present existing ball iterations as special cases. To find a zero of a quasi-strongly monotone mapping, which arises ofren 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 estima­ tion 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 com­ pletely 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.