On first zero crossing points

Abstract

Three algorithms, FZ1, FZ2 and FZ3 for bounding the first zero crossing point of a set of univariate functions on a bounded closed interval are described. Extended interval arithmetic is used in both FZ2 and FZ3. Automatic derivative arithmetic is used in FZ2 and automatic slope arithmetic is used in FZ3. Numerical results are presented.

Introduction

Let $\text{f}{}_{\mathrm{i}}\text{:D⊆}\text{R}{}^1\text{→}\text{R}{}^1$ (i=1,…,n) be given functions with f_i∈C(D). The so-called first zero crossing point (FZCP) problem consists of determining $\text{x}{}^{\mathrm{\ast }}\text{∈[a,b]⊆D}$ such that

$$\text{x}{}^{\mathrm{\ast }}\text{=}\text{min}{}_{\mathrm{1⩽i⩽n}}\text{{x}{}_{\mathrm{i}}{}^{\mathrm{\ast }}\text{},}$$

where for i=1,…,n, $\text{f}{}_{\mathrm{i}}\text{(x}{}_{\mathrm{i}}{}^{\mathrm{\ast }}\text{)=0}$, f_i(a)⩾0, and $\text{x}{}_{\mathrm{i}}{}^{\mathrm{\ast }}$ is the least zero of f_i in [a,b].

The FZCP problem has recently been addressed by Casado et al. [1] who have described a simple, elegant and efficient algorithm FZCP_IBB, for the case in which n=1 and when the functions f_i are not necessarily differentiable. The authors of FZCP_IBB have described, in [2] an algorithm, the so-called minimal root finder reordered algorithm for a set of n⩾1 functions, which will be referred to in this paper as CGS. The algorithms FZCP_IBB and CGS may be used to solve problems for which the functions f_i (i=1,…,n) are not necessarily differentiable (that is to say, the f_i might or might not be differentiable). However it is often required to solve the FZCP problem in situations in which f_i∈C¹(D) (i=1,…,n) (see, for example [3], [4], [5], [6]). As has been demonstrated in [3], interval arithmetic may be used effectively for ray intersection when the univariate function the zero of which is to be determined is continuously differentiable. Therefore it seems to be desirable to investigate the use of interval arithmetic for the solution of the FZCP problem when f_i∈C¹(D) (i=1,…,n).

The algorithm FZ1 is closely based on the algorithm CGS. These algorithms are intended to be used on functions which are not necessarily continuously differentiable, and they have been implemented in order to provide a comparison with the algorithm FZ2 in which automatic derivative arithmetic is used and a comparison with the algorithm FZ3 in which automatic slope arithmetic is used.

Section 2 contains the notation. Some details concerning the interval arithmetic and the interval differentiation and slope arithmetic which is used in the algorithms FZ1, FZ2 and FZ3 are given in Section 3. The algorithms FZ1, FZ2 and FZ3 are described in Section 4. Numerical results which are obtained from the algorithms FZ1, FZ2 and FZ3 are presented in Section 5. Section 6 provides some conclusions.