Engineering topics
Publications and source records attributed to Ng, E. W..
The NASA software, in broad categories, is described, and the software life cycle is characterized. Programming language policy and practices are reviewed. Applicability, benefits, and transition, as well as present and potential problems with Ada are examined.
The gravitational potential of a disk is given, as is the gravitational field at a point in space. It is pointed out that formulas for a ring can be obtained as the difference between the results presented here for two different values of the disk radius. Results are obtained in terms of elliptic integrals and it is shown how these functions can be computed efficiently. Formulas necessary for the computation of partial derivatives are also presented.
This paper describes one example of multiple Fourier series which originate from a problem of spectral analysis of time series data. The example is exercised here with an algorithmic approach which can be generalized for other series manipulation on a computer. The generalized approach is presently pursued towards applications to a variety of multiple series and towards a general purpose algorithm for computer algebra implementation.
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.
An efficient algorithm is presented for the solution of Kepler's equation f(E)=E-M-e sin E=0, where e is the eccentricity, M the mean anomaly and E the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics in M, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20,000 pairs of values of e and M show that for single precision, 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision one additional iteration is required.
A second order problem is outlined for which MACSYMA was used only for checking purpose. A fourth order problem is briefly described for which a symbolic system is necessary. At the end, some sample output is displayed to indicate the complexity of the computational problem.
A class of integration strategies is explored that fall in between the two extremes of symbolic integration and numerical quadrature, which are, respectively, aimed at the computer generation of answers in the form of exact expressions and numerical values. The theoretical advances in symbolic integration are discussed. Three major contexts of applications with attendant case studies, four possible types of strategies for approximate integration are examined. The feasibility and adequacy (or inadequancy) of MACSYMA for implementing these strategies are considered.
Linear transformations are well-known in the theory of hypergeometric functions. In this note, it is indicated, both by analyses and by supporting numerical experiments, how these transformations can be applied to the computation of Legendre's functions, the incomplete Beta function, and the variance-ratio probability distribution function. It is shown that a simple transformation can in many cases cause dramatic improvement in computation.
An algorithm is presented for the symbolic integration of a class of algebraic functions. This class consists of functions made up of rational expressions of an integration variable x and square roots of polynomials, trigonometric and hyperbolic functions of x. The algorithm is shown to consist of the following components:(1) the reduction of input integrands to conical form; (2) intermediate internal representations of integrals; (3) classification of outputs; and (4) reduction and simplification of outputs to well-known functions.
A survey and comparison of some computational methods and algorithms for gamma and log-gamma functions of complex arguments are presented. Methods and algorithms reported include Chebyshev approximations, Pade expansion and Stirling's asymptotic series. The comparison leads to the conclusion that Algorithm 421 published in the Communications of ACM by H. Kuki is the best program either for individual application or for the inclusion in subroutine libraries.
Integrals of products of error functions with other functions are presented, taking into account a combination of the error function with powers, a combination of the error function with exponentials and powers, a combination of the error function with exponentials of more complicated arguments, definite integrals from Laplace transforms, and a combination of the error function with trigonometric functions. Other integrals considered include a combination of the error function with logarithms and powers, a combination of two error functions, and a combination of the error function with other special functions.
Algorithm and subroutine for solving Kepler equation for elliptical orbits
Confluent hypergeometric functions indefinite integrals analytic expressions and reduction formulas
Table of definite and indefinite integrals of products of exponential integrals with elementary or transcendental functions
Chebyshev series approximations for Bose- Einstein functions of orders one to ten, noting computability of Fermi-Dirac functions and polylogarithms
Tabulation of definite and indefinite integrals of products of error function with elementary and transcendental functions