Regula-Falsi法による近似解の収束性について

Abstract

The regula-falsi method is very practical when we search for the approximate solutions of non-linear equation f(x) = 0. Under the additional conditions on the smoothness of f(x), we can verify that the approximation sequence obtained by the regula-falsi method converges to the true solurtion of f(x) = 0. But in the case that f(x) is not smooth, the convergence of the approximate sequence can't be proved in general. The main purpose of this paper is to obtain the following Theorem. Let f(x) be continuous on the closed interval [a, b]. Suppose that f(a)・f(b) < 0. Then both of the inferior and the superior limits of the approximate sequencee by the regula-falsi method are the true solutions of f(x) = 0. And we shall consider some applications of the above theorem.