Wetherell recently described an algebra of error values that could be added to the ordinary arithmetic of a programming language. Along with ordinary arithmetic values, error values were included in the set of computational quantities. The error values could participate in all arithmetic operations and return meaningful results. Unfortunately, the definitions of the error values were not precise enough. Using Brown's model of computer arithmetic, we supply precise definitions for the error values, define the fundamental arithmetic operations on the new values, comment on their properties, and discuss briefly how they might be used and implemented. We also compare our model to the error handling features of the proposed IEEE floating point standard.

A model of floating-point computation, intended as a basis for efficient portable mathematical software, is presented.The model involves only simple familiar concepts, expressed in a small set of environment parameters.Using these, both a program and its documentation can tailor themselves to the host computer.First, the model numbers, a conventional four-parameter system of floating-point numbers, are introduced.The parameters are the base, the precision, and the minimum and maximum exponents.They must be chosen so that the result of a basic arithmetic operation on model numbers is no less accurate than the result that would be obtained by chopped arithmetic in the model system.Also, the result of a basic operation on operands between model numbers must be bounded by the results of the same operation on the neighboring model numbers.These ideas are summarized in a few fundamental axioms, which enable the machine-independent properties of numerical programs to be stated and proved.The main conclusion is that the model supports conventional, worst case, forward or backward error analyses with only minor qualifications.The model is simple in the sense that its axioms are more easily stated and understood than the detailed operational rules of most floating-point processors.It is realistic in the sense that real computers exhibit most of the forms of behavior (or misbehavior) that it allows, but nearly always conform to its rules.

This paper presents a language-independent proposal for envtronment parameters and basic functions for floating-point computation, and suggests a speofic representation m terms of generic functions for Fortran 77 The enwronment parameters were introduced m 1967 by Forsythe and Moler, who attributed the essentmls of their theory to Wilkinson These parameters are also used m the PORT mathemaucal subroutine library, with precise definitions in terms of a more recent model of floatingpoint computatmn, and a similar set has been proposed by the IFIP Working Group on Numerical Software.Three of the basra functmns are taken from another proposal by this group, but redefined m terms of the parameters and the model to provide a firm theoretical foundation The other three basic functmns can be expressed in tetras of these, but we feel they should be provided separately for convenienceThe proposed parameters specify the base of the floating-point number system, the maximum precision consistent with the representational and operatmnal accuracy of the machine, and the maxnnum exponent range within whmh one can compute without fear of overflow, underflow, or unexpected loss of accuracy The proposed functions separate a floating-point number into its constituents, reconstruct a number from those constituents, scale a number by a power of the base, and determine the absolute or relatwe spacing (in the sense of the model) between numbers in the vmmlty of a given number.Key Words and Phrases' environment parameters,

article The Subresultant PRS Algorithm Share on Author: W. S. Brown Bell Laboratories, 600 Mountain Ave., Murray Hill, NJ Bell Laboratories, 600 Mountain Ave., Murray Hill, NJView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 4Issue 3Sept. 1978 pp 237–249https://doi.org/10.1145/355791.355795Online:01 September 1978Publication History 53citation574DownloadsMetricsTotal Citations53Total Downloads574Last 12 Months42Last 6 weeks5 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 AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm.The phenomenon of coefficient growth is described, and the history of successful efforts first to control it and then to eliminate it is related.The recently developed modular algorithm is presented in careful detail, with special attention to the case of multivariate polynomials.The computing times for the classical algorithm and for the modular algorithm are analyzed, and it is shown that the modular algorithm is markedly superior.In fact, in the multivariate ease, the maximum computing time for the modular algorithm is strictly dominated by the maximum computing time for the first pseudo-division in the classical algorithm.

This paper presents an elementary treatment of the theory of subresultants, and examines the relationship of the subresultants of a given pair of polynomials to their polynomial remainder sequence as determined by Euclid's algorithm.Two important versions of Euclid's algorithm are discussed.The results are essentially the same as those of Collins, but the presentation is briefer, simpler, and somewhat more general.

MERCURY is a computer-aided system for the selective distribution of Bell Telephone Laboratories technical reports to employees. MERCURY is based on the idea that the job of distribution should be divided between the author and the reader with each performing that part of the job which he does best. The author is asked to describe the interests of the readers to whom his report should be sent. The reader is asked to described the reports that he wishes to receive. The descriptions are made from a small structured vocabulary. The matching of report to reader and the printing of address labels is done by computer.