On first zero crossing points
Introduction
Let (i=1,…,n) be given functions with fi∈C(D). The so-called first zero crossing point (FZCP) problem consists of determining such thatwhere for i=1,…,n, , fi(a)⩾0, and is the least zero of fi 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 fi 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 fi (i=1,…,n) are not necessarily differentiable (that is to say, the fi might or might not be differentiable). However it is often required to solve the FZCP problem in situations in which fi∈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 fi∈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.
Section snippets
Notation
The fundamental interval mathematics that is required to understand this paper is described in [7], [8] (see also [9]). The notation that is needed is as follows. A real interval has infimum and supremum with . The set of real intervals is denoted by . If then . If and then one writes .
The midpoint , the magnitude , the width and the radius of are defined by
Using interval automatic derivative and slope arithmetic
The mathematical foundation of automatic derivative and slope arithmetic is given in [8]. Computational details are given in [10], [11], [12]. Formulae from which interval extensions of a continuously differentiable function , its first derivative and its slope may be determined are as follows.
Let be such that is the last in a finite sequence of expressions.
Let the derivative of be defined (if possible) by the following relations, in which
The algorithms FZ1, FZ2 and FZ3
In the algorithms FZ1, FZ2 and FZ3 which are described in this section, Q is a doubly linked list (initially empty) which ultimately contains intervals such that and L is an array of doubly linked lists the components Li of which contain subintervals of [a,b] which are to be processed to bound a solution of the FZCP problem if it exists. Initially Li={[a,b]} (i=1,…,n). Ultimately Li={∅} (i=1,…,n) and .
It is necessary, in FZ1, in FZ2 and in FZ3 to arrange that if
Numerical results
In order to compare the performance of the algorithms CGS, FZ1, FZ2 and FZ3 results for the determination of the FZCP of the 27 continuously differentiable functions in Table 1 taken from Table (3.1) in [2] are presented in this section. As pointed out in [2] the computational cost of CGS depends on the order in which the functions fi are evaluated; this has also been found to be so for FZ1, FZ2 and FZ3.
Results for the functions 1–27 are given in Table 2 and results for functions 27–1 are given
Conclusions
The computational costs of CGS and FZ1 appear to be identical, as too are the CPU times. This is interesting because in CGS n lists Qi (i=1,…,n) are used while in FZ1 only one list Q is used. For both CGS and FZ1 T=2×10−2 s, while for FZ2 and FZ3 T is effectively zero. Therefore it would appear that CGS and FZ1 are equally effective, but that FZ2 and FZ3 are quicker.
References (15)
- et al.
An algorithm for finding the zero crossing of time signals with Lipschitzean derivatives
Measurement
(1995) - et al.
Fast detection of the first zero-crossing in a measurement signal set
Measurement
(1996) Interval mathematics, algebraic equations and optimization
Journal of Computational and Applied Mathematics
(2000)Rigorous sensitivity analysis for parameter-dependent systems of equations
Journal of Mathematical Analysis and Applications
(1989)- et al.
Interval branch and bound algorithm for finding the first-zero-crossing-point in one-dimensional functions
Reliable Computing
(2000) - L.G. Casado, I.F. García, Ya.D. Sergeyev, Interval algorithms for finding the minimal root in a set of multiextremal...
- et al.
Robust and efficient ray intersection of implicit surfaces
Reliable Computing
(2000)
Cited by (2)
Acceleration of univariate global optimization algorithms working with lipschitz functions and lipschitz first derivatives
2013, SIAM Journal on OptimizationHigher order numerical differentiation on the Infinity Computer
2011, Optimization Letters