A Generalized False Position Numerical Method for Finding Zeros and Extrema of a Real Function

摘要

Root-finding numerical methods are based on the Intermediate Value Theorem. It states that a root of a real function f : R → R is bracketed in a given interval [A, B] is contained in R if f(A) and f(B) have opposite signs, i.e. f(A).f(B) < 0. But, some roots, say local minima or maxima, cannot be bracketed this way because the condition f(A).f(B) < 0 is not satisfied. In this case, we normally use a specific numerical method for bracketing an extremum, checking then whether it is a zero of f or not. In contrast; this paper introduces a single numerical method, called generalised false position method, that is capable of computing not only zeros but also extrema of a function. Thus, it determines any zero in a given interval [A, B] is contained in R even when the Intermediate Value Theorem is not satisfied. This is particularly important in sampling implicit curves and surfaces that evaluate either positive or negative everywhere, a computer graphics problem that has remained unsolved for so long.