Ultra-arithmetic I: Function data types

Abstract

We develop ultra-arithmetic, a calculus for functions which is performable on a digital computer. We proceed with an analogy between the real numbers and their truncated positional number representations on the one hand and functions and their truncated generalized-Fourier series on the other. Thus we lift the digital computer from a setting corresponding to the real numbers to a setting corresponding to function spaces. Digitized function data types are defined along with computer versions of the corresponding ultra-arithmetic operations (addition, subtraction, multiplication, division, differentiation and integration). Error estimates for these operations (the analogues of traditional rounding errors) are given. Explicit examples of the error estimates for the ultra-arithmetic operations are given in the cases of five specific choices of basis functions; Fourier-, Chebyshev-, Legendre-, sine- and cosine-bases. Finally the algorithms of ultra-arithmetic are given in an explicitly implementable form for the cases both of the Fourier basis and the Chebyshev basis. © 1982.