Algorithm of the Bi-Month: Elementary Transcendental Functions

This chapter describes floating-point mathematics for real-time C++ using built-in floating-point types such as Open image in new window and Open image in new window . The first sections of this chapter introduce floating-point arithmetic, mathematical constants, elementary transcendental functions and higher transcendental functions. The last sections of this chapter cover more advanced topics including complex-numbered mathematics, compile-time evaluation of floating-point functions and generic numeric programming.

* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

In this paper using a perturbative method a series expansion of the price of a (put up and out) barrier option with time dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and non elementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature and a sufficient condition for the convergence of the perturbation series expansion is given. The numerical experience shows that in the cases of practical interest considered the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options such as step options, step barrier options and so on. The website contains some auxiliary material that helps the understanding of this paper and the computer programs needed to evaluate the formula obtained.

1 - Variables and Functions

For the statistical description of a demand process, many probability density functions (pdf) are available, each having its advantages and drawbacks. How do they compare when in application? Is it true that the shape of the pdf is not relevant? This paper deals with one application, namely a simple s,Q stock control system. Assuming for the lead time demand successively five different pdf's (Gaussian, logistic, gamma, log-normal, Weibull) and using the service level as a criterion, it is shown that for this application (1) the numerical differences between the selected functions are in fact negligible; (2) the logistic pdf produces a formula that only contains elementary transcendental functions; (3) the other pdf's lead to much more complex expressions.

Over the past 24 years, the gravity fields of homogeneous polyhedra have been studied extensively (Paul, 1974; Barnett, 1976; Okabe, 1979; Golizdra, 1981; Strakhov et al., 1986a, 1986b; Gotze and Lahmeyer, 1988; Pohanka, 1988; Ivan, 1990; Pohanka, 1990; Holstein and Ketteridge, 1996). This intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run‐time library for most computational programming languages.

The parabolic equation (PE) approximation—essential in solving range‐dependent propagation problem—relies on iterated fast Fourier transforms (FFTs). Preventing significant errors due to the periodicity of the finite FFT (“ringing”) requires inserting an absorber potential to confine the wave energy to the range of interest. Such potentials, or sponges, have, to date, been strictly ad hoc. The problem of creating an optimal sponge by using an optical model analogy of the reflectionless potential U = U0* [sech(a*x)]2 of 1‐D quantum mechanics is studied. This approach permits the development of closed‐form expressions for both transmission and reflection coefficients in terms of elementary transcendental and gamma functions of complex argument. Using these coefficients and an efficient technique for evaluating such gamma functions yields a quantitative study of the problem of the optimal sponge. These results demonstrate that, for relevant parameter domains, there may be several “branches” of minimal paths ...

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

A single dendritic neuron model (DNM) that owns the nonlinear information processing ability of dendrites has been widely used for classification and prediction. Complex-valued neural networks that consist of a number of multiple/deep-layer McCulloch-Pitts neurons have achieved great successes so far since neural computing was utilized for signal processing. Yet no complex value representations appear in single neuron architectures. In this article, we first extend DNM from a real-value domain to a complex-valued one. Performance of complex-valued DNM (CDNM) is evaluated through a complex XOR problem, a non-minimum phase equalization problem, and a real-world wind prediction task. Also, a comparative analysis on a set of elementary transcendental functions as an activation function is implemented and preparatory experiments are carried out for determining hyperparameters. The experimental results indicate that the proposed CDNM significantly outperforms real-valued DNM, complex-valued multi-layer perceptron, and other complex-valued neuron models.