Algorithms have been developed to compute short- and long-precision elementary functions: SIN, COS, TAN, COTAN, LOG, LOG10, EXP, POWER, SQRT, ATAN, ASIN, ACOS, ATAN2, and CABS, in scalar (28 functions) and vector (22 functions) mode. They have been implemented as part of the new VS FORTRAN library recently announced along with the IBM 3090 Vector Facility. These algorithms are essentially table-based algorithms. An important feature of these algorithms is that they produce bitwise-identical results on scalar and vector System/370 machines. Of these, for five functions the computed value result is always the correctly rounded value of the infinite-precision result. For the rest of the functions, the value returned is one of the two floating-point neighbors bordering the infinite-precision result, which implies exact results if they are machine-representable. For the five correctly rounded elementary functions, scalar and vector algorithms are designed independently so as to optimize performance in each case. For other functions, the bitwise-identical constraint leads to algorithms which compromise between scalar and vector performance. The authors have been able to design algorithms where this compromise is minimal and thus achieve very good performance on both scalar and vector implementations. For test measurements on high-end System/370 machines, scalar functions aremore » always faster (sometimes by as much as a factor of 2.5) as compared to the old VS FORTRAN library. The vector functions are usually two to three times faster than scalar functions.« less

New scalar and vector elementary functions for the IBM system/370

Chebyshev approximation by ordinary rational functions with respect to a vanishing weight function is considered. A best approximation is characterized by alternation and is unique but may not exist. The problem arises in Kuki's technique for rational approximation with interpolation at zero and with Williams' interpolating rationals. Let [a, b] be a finite interval and C[a, b] be the set of continuous functions on [a, b] . For g E C[a, b] define 11ghl = max {Ig(x)l: a 0 for a 1, the theory of approximation by generalized rational functions, as developed by Boehm and Goldstein [6, pp. 84-89], guarantees that there is a ratio r = p/q such that Ils(f r) II is minimal, but such an r may have a pole in [a, b]. Example. Let [a, b] = [0,1 ],n = 0, m = 1,s(x) =x, and f(x) = 2, 0 < x < 1/2, = 1/x + 4 sin(67rx)/x, 1/2 < x < 1. Let ro(x) = 1/x; then Ils(f ro)II = 4 and s(f ro) alternates twice on [0, 1] with extrema at 7/12, 9/12, 11/12. From this it follows by standard arguments in rational approximation that ro is the unique ratio of polynomials of degree 0 to polynomials of degree 1 which minimizes Ils(fr)II. We have Ils(f1/(1/k + x))I l 4; so no best approximation in R ? [0, 1 ] exists. It should be noted that Taylor and Williams [8] have related examples of nonexistence in Williams' problem. Received April 24, 1975. AMS (MOS) subject classifications (1970). Primary 41A50. Copyright ? 1976, American Mathematical Society

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Rational approximation with a vanishing weight function and with a fixed value at zero

Bibliography on the evaluation of numerical software

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1320&context=cstech

An analytical study of strategy-oriented restructuring algorithms☆

We review software and hardware of micro-computers in the light of requirements for the production of robust software. We consider that attitudes about such production on large scale computers which have been developed over the last fifteen years should be adhered to, and that user protection from inadequate software is most important. The typical computer user will tend more and more to consider computational ingredients made available to him to be "black boxes", and will be prone to rely on results in the same manner as he relies on arithmetic componentry to perform correct operations.

Scientific computations using micro-computers

This paper discusses a FORTRAN subprogram library developed primarily to support extended-precision floating-point arithmetic. The general strategy, which makes limited use of guard digits, is developed to achieve high accuracy with reasonable execution time and storage space.

In addition to describing some previously unpublished algorithms, the authors present subprograms for simulating extended-precision arithmetic and for input and output conversion.

J. Ascoly

Papers

FORTRAN extended-precision library