A generalized regula falsi method for finding zeros and extrema of real functions

Abstract

Many zero-finding numerical methods are based on the Intermediate Value Theorem, which states that a zero of a real function f: R → R is bracketed in a given interval A, B ⊂ R if f A and f B have opposite signs; that is, f A · f B < 0. But, some zeros cannot be bracketed this way because they do not satisfy the precondition f A · f B < 0. For example, local minima and maxima that annihilate f may not be bracketed by the Intermediate Value Theorem. In this case, we can always use a numerical method for bracketing extrema, checking then whether it is a zero of f or not. Instead, this paper introduces a single numerical method, called generalized regula falsi (GRF) method to determine both zeros and extrema of a function. Consequently, it differs from the standard regula falsi method in that it is capable of finding any function zero in a given interval A, B ⊂ R even when the Intermediate Value Theorem is not satisfied. © 2013 Abel Gomes and José Morgado.