Chains of recurrences for functions of two variables and their application to surface plotting

Abstract

When generating curves or surfaces of closed-form mathematical functions, usually the most time-consuming task is function evaluation at discrete points. Most programs (among them most of the existing computer algebra systems) achieve this by straightforward evaluations of linearly sampled points through whatever numerical evaluation routines the particular system provides. More specifically, most programs use evaluations of the following form:

$$G\left( {{{x}_{0}} + n{{h}_{x}},{{y}_{0}} + m{{h}_{y}}} \right) for all n = 0, \ldots ,N,m = 0, \ldots ,M $$

for some given two-dimensional function G(x, y), starting points x ₀, y ₀ and increments h _x, h _y. For example, the following loop is used inside Maple’s plot 3d function (Char et al. 1988):

$$\begin{gathered} xinc : = \left( {xmax - xmin} \right)/m; yinc : = \left( {ymax - ymin} \right)/n; x: = xmin; \hfill \\ for i from 0 to m do \hfill \\ y : = ymin; \hfill \\ for j from 0 to n do z\left[ {i,j} \right] : = f\left( {x,y} \right); y : = y + yinc od; \hfill \\ x : = x + xinc \hfill \\ od; \hfill \\ \end{gathered} $$