A Generalized Regula Falsi Method for Finding Zeros and Extrema of Real Functions

摘要

Many zero-finding numerical methods are based on the Intermediate Value Theorem, which states that a zero of a real functionf:ℝ→ℝis bracketed in a given intervalA,B⊂ℝiffAandfBhave opposite signs; that is,fA·fB<0. But, some zeros cannot be bracketed this way because they do not satisfy the preconditionfA·fB<0. For example, local minima and maxima that annihilatefmay 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 offor not. Instead, this paper introduces a single numerical method, calledgeneralized regula falsi(GRF) method to determine both zeros and extrema of a function. Consequently, it differs from the standardregula falsi methodin that it is capable of finding any function zero in a given intervalA,B⊂ℝeven when the Intermediate Value Theorem is not satisfied.