Showing papers on "Elementary function published in 1993"

Linear forms in the logarithms of algebraic numbers

Abstract: This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter. *** DIRECT SUPPORT *** A00I6B17 00002

BKM: A new hardware algorithm for complex elementary functions

Abstract: An algorithm for computing complex logarithms and exponentials is proposed. The algorithm is based on shift-and-add elementary steps, and it generalizes the Cordic algorithm. It can compute the usual real elementary functions. This algorithm is more suitable for computations in a redundant number system than Cordic, since there is no scaling factor for computation of trigonometric functions. >

On the potentials of galactic discs

Abstract: The standard Bessel function formula for the potential of a thin axisymmetric disc is extended to include arbitrary vertical structure, yielding a formula which reduces the potential to a single quadrature in the important cases of a disc with constant scale-height and either exponential or Gaussian vertical density profiles. The general solution of Poisson's equation for an axisymmetric body is also given as a double integral over a Legendre function. A new formulation for the potential of thin axisymmetric discs is also given. The potential is expressed as a double integral over elementary functions in the most general case, and can usually be reduced to a single quadrature

Algorithm 714: CELEFUNT: a portable test package for complex elementary functions

Abstract: This paper discusses CELEFUNT, a pakage of Fortran programs for testing complex elementary functions.

Approximation of the Hersch-Pfluger distortion function.

Abstract: This paper aims at giving a method of approximating the distortion function ΦK by means of elementary functions As a consequence, new bounds for the function ΦK are established Moreover, the n -dimensional counterpart ΦK,n of ΦK , n = 2, 3, is considered

A note on a class of spherically symmetric solutions

Abstract: We find the general solution of the Einstein field equations in terms of elementary functions for a class of accelerating, expanding and shearing spherically symmetric metrics. We demonstrate that these solutions satisfy the equation of statep = π+ const, which is a generalisation of the stiff equation of state. The properties of the solutions are briefly discussed; in particular we show that our solutions have constant anisotropy. We relate our results to the solutions of Gutman and Bespal’ko, Wesson, Lake, Shaver and Lake, and Hajj-Boutros. We show that the Wesson solution is not new and is equivalent to the metric found earlier by Gutman and Bespal’ko. This equivalence was also noted by Lake. We explicitly find the coordinate transformation that equates these solutions. We also show that solutions given by Hajj-Boutros, for particular choices of a metric function, in terms of Painleve transcendents can be written completely in terms of elementary functions. This is consistent with the results of Shaver and Lake who have shown the equivalence of the Hajj-Boutros and Lake metrics.

The approximate evaluation of certain integrals

Abstract: An adaptation of the decomposition method allows one to calculate an integral not expressible in terms of elementary functions nor adequately tabulated. It is shown that only two terms of the series solution lead to an excellent agreement with the actual solution of the problem.

Linear Dynamical Systems

Abstract: Linear dynamical systems (whether they be linear differential equations with constant coefficients or iterative systems) are the only important class of higher dimensional systems that can be solved in terms of elementary functions.

Comments on "Computation of the circular error probability integral" by J.T. Gillis

Abstract: A numerical technique for computing the circular error probability (CEP) integral using only elementary functions is presented. It is simpler and more accurate than the truncated series solution in the above-titled paper (see ibid., vol.27, p.906-10, Nov. 1991). >

A fourth-order solution of the ideal resonance problem

Abstract: The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a ‘Kepler-equation’. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians.

Dynamic Forms. Part 1: Functions

Abstract: The formalism of dynamic forms is developed as a means for organizing and systematizing the design control systems. The formalism allows the designer to easily compute derivatives to various orders of large composite functions that occur in flight-control design. Such functions involve many function-of-a-function calls that may be nested to many levels. The component functions may be multiaxis, nonlinear, and they may include rotation transformations. A dynamic form is defined as a variable together with its time derivatives up to some fixed but arbitrary order. The variable may be a scalar, a vector, a matrix, a direction cosine matrix, Euler angles, or Euler parameters. Algorithms for standard elementary functions and operations of scalar dynamic forms are developed first. Then vector and matrix operations and transformations between parameterization of rotations are developed in the next level in the hierarchy. Commonly occurring algorithms in control-system design, including inversion of pure feedback systems, are developed in the third level. A large-angle, three-axis attitude servo and other examples are included to illustrate the effectiveness of the developed formalism. All algorithms were implemented in FORTRAN code. Practical experience shows that the proposed formalism may significantly improve the productivity of the design and coding process.

Design of a VLSI circuit for on-line evaluation of several elementary functions using their Taylor expansions

Abstract: The authors present a new modular architecture for the online evaluation of power series. It can be used to quickly compute any function that can be approximated by the first terms of its Taylor expansion (i.e., most math functions). For trigonometric functions, the method matches Cordic-like methods and leads to more regular architectures. The authors also present a VLSI implementation of this architecture. >

Chapter VI – Fractal Interpolation

Abstract: Publisher Summary This chapter discusses fractal interpolation functions. The fractal dimension of the graphs of Euclidean functions is always 1; these elementary Euclidean functions are useful not only because of their geometrical content but also because they can be expressed by simple formulas. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms. The graphs of these functions can be used to approximate image components; rather than treating the image component as arising from a random assemblage of objects, it can be well defined using fractal interpolation functions. Fractal interpolation functions also provide a new means for fitting experimental data. Fractal interpolation functions share with elementary functions that they are of a geometrical character, that they can be represented succinctly by formulas, and that they can be computed rapidly. The main difference is their fractal character.

Selecting the projection functions used in an iterative Gabor expansion

Abstract: This paper discusses the selection of projection functions used in an iterative implementation of the Gabor expansion. We show that the optimal support-limited projection function corresponds to a truncated version of Bastiaans' biorthonormal projection function for the case of a harmonic lattice. For various support widths, the lower bound of the optimal convergence factor is calculated. It is shown that Gabor's original projection function, which corresponds to the central lobe of Bastiaans' biorthonormal projection function, is truncated too severely, producing a significant overlap with elementary functions from high frequency channels. As a result, the lower bound for the optimal convergence factor and the rate of convergence will approach zero as the signal bandwidth (and the highest frequency Gabor channel) is increased. This work also determines the lower bound of the optimal convergence factor for projection functions implemented using log-polar lattices. For both the harmonic and log-polar lattices, we investigate the trade-off between spread of convergence and the size of the projection function.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.