Computer Generation of Optimized Subroutines

Abstract

Ideally, subroutines used for function evaluation should be both flexible (suitable for use over arbitrary intervals as well as accurate to any desired order) and fast. Often these characteristics conflict, and, as an alternative, a number of subroutines for the same function may be kept in the Subroutine Library, with the programmer selecting the one closest to his needs. Among the fastest subroutines are those using fixed-degree polynomial approximations, where the fixed number of terms has been selected so that the error does not exceed a certain limit throughout the interval. This paper describes a subroutine generator program which, for a number of functions, produces a fixed-degree polynomial approximation subroutine which is adapted to the particular interval and accuracy needed by the programmer (within wide limits). The generation of the polynomial is based on the use of Chebyshev polynomials, and therefore the approximation is optimum in the sense of the Chebyshev polynomial approximations (for a given degree polynomial, very nearly minimum maximum error over the interval).