We develop efficient algorithms for reliable and accurate evaluatins of the complex arcsine and arccosine functions. A tight error bound is derived for each algorithm; the results are valid for all machine-representable points in the complex plane. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. Corresponding Fortran 77 programs for an IEEE environment have also been developed to illustrate the practicality of the algorithms, and these programs have been tested very carefully to help confirm the correctness of the algorithms and their error bounds. The results of these tests are included in the article, but the Fortran 77 programs are not (these programs are available from Fairgrieve). Tests of other widely available programs fail at many points in the complex plane, and otherwise are slower and produce much less accurate results.

Formulas developed originally by Weierstrass have been used since the 1960s by many others for the simultaneous determination of all the roots of a polynomial. Convergence to simple roots is quadratic, but individual approximations to a multiple root converge only linearly. However, it is shown here that the mean of such individual approximations converges quadratically to that root. This result, along with some detail about the behavior of such approximations in the neighborhood of the multiple root, suggests a new approach to the design of polynomial rootfinders. It should also be noted that the technique is well suited to take advantage of a parallel environment. This article first provides the relevant mathematical results: a short derivation of the formulas, convergence proofs, an indication of the behavior near a multiple root, and some error bounds. It then provides the outline of an algorithm based on these results, along with some graphical and numerical results to illustrate the major theoretical points. Finally, a new program based on this algorithm, but with a more efficient way of choosing starting values, is described and then compared with corresponding programs from IMSL and NAG with good results. This program is available from Mathon ([email protected]).

Algorithms are developed for reliable and accurate evaluations of the complex elementary functions required in FORTRAN 77 and FORTRAN 9, namely, cabs, csqrt, cexp, clog, csin, and ccos. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. A tight error bound is derived for each algorithm. Corresponding FORTRAN programs for an IEEE environment have also been developed to illustrate the practicality of the algorithms, and these programs have been tested very carefully to help confirm the correctness of the algorithms and their error bounds. The results of these tests are included in the paper, but the FORTRAN programs are not.

The authors have been developing a programming system intended to be especially convenient for scientific computing. Its main features are variable precision (decimal) floating-point arithmetic and convenient exception handling. The software implementation of the system has evolved over a number of years, and a partial hardware implementation of the arithmetic itself was constructed and used during the early stages of the project. Based on this experience, the authors have developed a set of specifications for an arithmetic coprocessor to support such a system. These specifications are described. An outline of the language features and how they can be used is also provided, to help justify the particular choice of coprocessor specifications. The authors also indicate what other hardware features would be most helpful to the systems programmer, especially for implementation of the exception handling.>

This paper describes some simple programming language facilities for raising and handling exceptions, and demonstrates, with a number of examples, their effectiveness in a scientific computing environment. Only a small number of predefined exceptions need to be introduced into the language, but any number of user-defined exceptions can he introduced by the programmer. A key feature is that handlers are attached to operators that can raise exceptions (including user-defined functions and procedures, as well as built-in operators), rather than to expressions or statements. The programmer has a great deal of freedom in specifying what action is to be taken in case an exception arises. The implementation and the applications are mainly concerned with scientific computing, but we believe the facilities are applicable in other areas as well.

A new computer arithmetic is described. Closely related built-in functions are included. A user's point of view is taken, so that the emphasis is on what language features are available to a user. The main new feature is flexible precision control of decimal floating-point arithmetic. It is intended that the language facilities be sufficient for describing numerical processes one might want to implement, while at the same time being simple to use, and implementable in a reasonably efficient manner. Illustrative examples are based on experience with an existing software implementation.

The exponential function presented here returns a result which differs from e x by less than one unit in the last place, for any representable value of x which is not too close to values for which e x would overflow or underflow. (For values of x which are not within this range, an error condition is raised.) It is a “variable precision” function in that it returns a p -digit approximation for a p -digit argument, for any p = 0 ( p -digit means p -decimal-digit). The program and analysis are valid for all p = 0, but current implementations place a restriction on p . The program is presented in a Pascal-like programming language called Numerical Turing which has special facilities for scientific computing, including precision control, directed roundings, and built-in functions for getting and setting exponents.

The square root function presented here returns a properly rounded approximation to the square root of its argument, or it raises an error condition if the argument is negative. Properly rounded means rounded to the nearest, or to nearest even in case of a tie. It is variable precision in that it is designed to return a p -digit approximation to a p -digit argument, for any p > 0. (Precision p means p decimal digits.) The program and the analysis are valid for all p > 0, but current implementations place some restrictions on p .

This paper describes the design of an arithmetic unit called CADAC (clean arithmetic with decimal base and controlled precision). Programming language specifications for carrying out "ideal" floating-point arithmetic are described first. These specifications include detailed requirements for dynamic precision control and exception handling, along with both complex and interval arithmetic at the level of a programming language such as Fortran or PL/I.

The purpose of this paper is to summarize proposed specifications for floating-point arithmetic and elementary functions. The topics considered are: the base of the number system, precision control, number representation, arithmetic operations, mother basic operations, elementary functions, and exception handling. The possibility of doing without fixed-point arithmetic is also discussed. The specifications are intended to be entirely at the level of a programming language such as Fortran. The emphasis is on convenience and simplicity from the user's point of view. The specifications are not complete in every detail, but it is intended that they be complete "in spirit" - some further details, especially syntatic details, would have to be provided, but the proposals are otherwise relatively complete.

Random number generators of the mixed congruential type have recently been proposed. They appear to have some advantages over those of the multiplicative type, except that their statistical behavior is unsatisfactory in some cases. It is shown theoretically that a certain class of these mixed generators should be expected to fail statistical tests for randomness. Extensive testing confirms this hypothesis and makes possible a more precise definition of the unsatisfactory class. It is concluded that the advantages of mixed generators can be realized only in special circumstances. On machines with relatively short multiplication times the multiplicative generators are to be preferred.

article Free Access Share on Mixed Congruential Random Number Generators for Decimal Machines Authors: J. L. Allard University of British Columbia, Vancouver, B.C., Canada University of British Columbia, Vancouver, B.C., CanadaView Profile , A. R. Dobell University of British Columbia, Vancouver, B.C., Canada University of British Columbia, Vancouver, B.C., CanadaView Profile , T. E. Hull University of British Columbia, Vancouver, B.C., Canada University of British Columbia, Vancouver, B.C., CanadaView Profile Authors Info & Claims Journal of the ACMVolume 10Issue 2April 1963 pp 131–141https://doi.org/10.1145/321160.321163Online:01 April 1963Publication History 21citation423DownloadsMetricsTotal Citations21Total Downloads423Last 12 Months23Last 6 weeks3 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

article Share on Efficiency of Predictor-Corrector Procedures Authors: T. E. Hull The University of British Columbia, Vancouver, B.C., Canada The University of British Columbia, Vancouver, B.C., CanadaView Profile , A. L. Creemer Shell Oil Company, Calgary, Alberta, Canada and The University of British Columbia, Vancouver, B.C., Canada Shell Oil Company, Calgary, Alberta, Canada and The University of British Columbia, Vancouver, B.C., CanadaView Profile Authors Info & Claims Journal of the ACMVolume 10Issue 301 July 1963pp 291–301https://doi.org/10.1145/321172.321176Published:01 July 1963Publication History 36citation431DownloadsMetricsTotal Citations36Total Downloads431Last 12 Months11Last 6 weeks0 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 SiteeReaderPDF