An implementation of automatic array arithmetic by a generalized push-down stack

Abstract

One of the most fundamental and useful notions of mathematical analysis is the concept of a continuous, single valued function of one independent variable. By y = f(x) we mean that for every x in the range of x, defined as α ≤ x ≤ β, the mapping f provides us with a value in the domain of the function y_α ≤ y ≤ y_β, where y_α = f(x_α) and y_β = f(x_β). In a numerical computation we represent the part of the range of the independent variable, which is of interest to us, by a suitably chosen ordered set of n + 1 values (x₀, x₁, x₂, . . ., x_n), and a representation of any function over this range is then found by evaluating y = f(x) at these points, to obtain a corresponding ordered set of values (y₀, y₁, y₂, . . ., y_n). Because of the obvious analogy, these arrays of numbers representing continuous functions are often called vectors. However, a semantic problem arises when we discuss vectors of functions, such as the vector potential or the wind velocity patterns in the atmosphere. Therefore, I prefer to make a special case of the representations of continuous functions and refer to them as arrays rather than vectors.