Daniel W. Lozier

NIST Digital Library of Mathematical Functions

The National Institute of Standards and Technology is preparing a Digital Library of Mathematical Functions (DLMF) to provide useful data about special functions for a wide audience. The initial products will be a published handbook and companion Web site, both scheduled for completion in 2003. More than 50 mathematicians, physicists and computer scientists from around the world are participating in the work. The data to be covered include mathematical formulas, graphs, references, methods of computation, and links to software. Special features of the Web site include 3D interactive graphics and an equation search capability. The information technology tools that are being used are, of necessity, ones that are widely available now, even though better tools are in active development. For example, LaTeX files are being used as the common source for both the handbook and the Web site. This is the technology of choice for presentation of mathematics in print but it is not well suited to equation search, for example, or for input to computer algebra systems. These and other problems, and some partially successful work-arounds, are discussed in this paper and in the companion paper by Miller and Youssef.

A Portable Extended Precision Arithmetic Package and Library with Fortran Precompiler

A set of ANS Fortran subroutines, developed by W.T. Wyatt, which incorporates computerindependent algorithms for performing arithmetic on arbitrary length, extended precision, floating point, real, or complex numbers in any user-designated eomputatmn base from 2 to 16 is described Also included is a hbrary of the Foi tran intrinsic and external (mathematical) functmns, as well as routines for type conversion, base conversion, input/output, and other operations. Analysis and testing, by D.W. Lozier, of the arithmetic and hbrary softwaie to locate and ehminate errors and to determine approximate accuracy hmlts is described. Finally, a precompfler, written in Fortran by D J. Orser, whmh translates a Fortran program containing two extended precision data types (SUPER PRECISION and SUPER COMPLEX) into an ordinary Fortran program that uses subroutine calls to the software to carry out the extended precision operatmns is described The software IS mtended to furmsh working scmnt~sts who are not specialists m computer science a ready means of computing m extended precision

Closure and precision in level-index arithmetic

First it is proved that two recently introduced systems of computer arithmetic, the level-index (li) and symmetric level-index (sli) systems are closed under the four basic arithmetic operations, provided that division by zero is excluded and the operations are executed in finite precision. In consequence, the li and sli systems are free from the defects of overflow and underflow.Second, measures of precision are discussed and compared. Third, the ranges and local precisions of numbers stored in the li and sli systems are compared with corresponding ranges and local precisions of numbers stored in the floating-point system.

Algorithms and codes for the Macdonald function: recent progress and comparisons

The modified Bessel function Kiv(x), also known as the Macdonald function, finds application in the Kontorovich-Lebedev integral transform when x and v are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared.

Software needs in special functions

Abstract Currently available software for special functions exhibits gaps and defects in comparison to the needs of moderm high-performance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for user-oriented testing software.

Generalized exponential and logarithmic functions

Abstract Generalizations of the exponential and logarithmic functions are defined and an investigation is made of two possible versions of these functions. Some applications are described, including computer arithmetic, properties of very large and very small numbers, and the determination of functional roots.

Computation of complex Airy functions and their zeros using asymptotics and the differential equation

We describe a method by which one can compute the solutions of Airys differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

Diffusion-Controlled Reaction in a Vortex Field

Abstract A two-dimensional model of a constant-density diffusion-controlled reaction between unmixed species initially occupying adjacent half-spaces is formulated and analyzed. An axisymmetric viscous vortex field satisfying the Navies-Stokes equations winds up thc interface between the species as they diffuse together and react. A flame-sheet approximation of the rapid reaction is made using a mixture fraction dependent variable. The problem was originally proposed by F. Marble, who performed a local analysis and determined the total consumption rate along thc flame sheet. The present paper describes a global similarity solution to the problem which is Fourier analyzed in a Lagrangian coordinate system. The Fourier amplitudes are determined both by an asymptotic analysis, valid for large Schmidt numbers,and by numerical solution of the two-point boundary-value ordinary differential equations. The solution is evaluated in both Lagrangian and Eulerian coordinate systems. Comparisons are made between the a...

A proposed software test service for special functions

This is a proposal to develop a software test service at the National Institute of Standards and Technology for use in testing the accuracy, or numerical precision, of mathematical software for special functions. The service would use the World Wide Web to receive test requests and return test results. The tests would be run on a network of workstations at the Institute. It is hoped that such a service will be of practical utility to anyone who uses special functions in physics or other applications, and that it will stimulate the interest of applied mathematicians who are interested in the computation of special functions as well as computer scientists who are interested in innovative uses of the Internet. The author solicits comments on any aspect of the proposed service.

Algorithm 567: Extended-Range Arithmetic and Normalized Legendre Polynomials [A1], [C1]

This algori thm consists of two logically distinct parts: (1) a package of six F O R T R A N subroutines to facilitate the use of a special form of computer floating-point ar i thmetic tha t we call extended-range arithmetic; and (2) a FORT R A N subroutine tha t computes values of normalized Legendre polynomials according to an algori thm tha t generates (for some inputs) floating-point numbers tha t are outside the range of any computer . Our desire to produce a robust F O R T R A N subroutine to compute these polynomials st imulated the development of the extended-range software package. This package may prove to be useful for many other computations. Normalized Legendre polynomials are defined by the formula