We describe a method by which one can compute the solutions of Airy's 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.

Floating-point underflow is often regarded as either harmless or as an indication that the computational algorithm is in need of scaling. A counterexample to this view is given of a function for which contour plotting is difficult due to floating-point underflow. The function arose as an asymptotic solution to a model problem in turbulent combustion in which two chemical species (fuel and oxidizer) mix and react in a vortex field. Scaling is not a viable option because of extreme sensitivity to a small physical parameter. Standard graphics software packages produce erroneous contours without any indication of difficulty. This example provides support for considering symmetric level-index arithmetic, a new form of computer arithmetic which is immune to underflow and overflow.>

Three forms of interval floating-point arithmetic are defined in terms of absolute precision, relative precision, and combined absolute and relative precision. The absolute-precision form corresponds to the centered form of conventional rounded-interval arithmetic. The three forms are compared on the basis of the number of floating-point operations needed to generate error bounds for inner-product accumulation.

This algorithm 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 arithmetic that we call extended-range arithmetic; and (2) a FOR-T R A N subroutine that computes values of normalized Legendre polynomials according to an algorithm that generates (for some inputs) floating-point numbers that 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 stimulated 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 ~) (1 -x~) ~/2 d P.(x),where ~ and ~ are nonnegative integers, x is a real variable lying in the closed interval [ -1 , 1], and P,(x) is the ordinary Legendre polynomial.These functions satisfy a three-term recurrence relation in ~ that is useful in c o m p u t i n g / ~( x ) for fixed ~ and x, and sequences #1, ~1 + 1 , . . ., #2 of values of #.For stability reasons, the recurrence is applied in the backward direction, starting at # = r + 1 and # -v and proceeding through ~ -1, p -2 . . . . .#1. T h e starting value of P~(x) is determined from a first-order recurrence relation, and P;+l(x) = 0 for all r, x.When x is close to =i:l and r is moderately large, this method fails because of Recewed

Nahonal Bureau of StandardsAn algorithm is presented for the computation of normalrzed Legendre polynomials.In order to permit wide ranges of argument, degree, and order of these functions, an "extended-range" arithmetic is introduced whereby a separate storage location is allocated to the exponent of a floatmg-pomt number.Since this device may have other applications, separate subroutines are developed for addition of extended-range numbers and also for conversion to and from ordinary floating-point form.

article Free Access Share on A Portable Extended Precision Arithmetic Package and Library with Fortran Precompiler Authors: W. T. Wyatt Harry, Diamond Laboratories, Adelphi, MD Harry, Diamond Laboratories, Adelphi, MDView Profile , D. W. Lozier Mathematical Analysis Section, Institute Basic Standards, National Bureau of Standards, Washington, DC Mathematical Analysis Section, Institute Basic Standards, National Bureau of Standards, Washington, DCView Profile , D. J. Orser Mathematical Analysis Section, Institute Basic Standards, National Bureau of Standards, Washington, DC Mathematical Analysis Section, Institute Basic Standards, National Bureau of Standards, Washington, DCView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 2Issue 3Sept. 1976 pp 209–231https://doi.org/10.1145/355694.355695Published:01 September 1976Publication History 22citation422DownloadsMetricsTotal Citations22Total Downloads422Last 12 Months7Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In view of the increasingly important role of the computer in scientific calculations, the development of computer algorithms for elementary and special functions has been given a great deal of attention. The development of algorithms cannot be divorced from their evaluation, for a computer algorithm is judged solely on the basis of its performance characteristics. These include storage requirements, speed and accuracy. The present paper will deal only with the accuracy aspect of algorithm testing. The other two aspects must be evaluated in the context in which the algorithm is used. In this paper by an algorithm we mean a computer algorithm, i.e. an implementation of a mathematical algorithm in a specific environment. The environment is taken to include factors that may affect the algorithm, e.g. the operating system under which the program is run and hardware algorithms for arithmetic operations. Whereas in some instances mathematical algorithm have been successfully used to locate hardware malfunctions that were not traceable by normal trouble shooting tests, any malfunctions of the software or hardware will not be considered here to be part of the environment.