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

  • Olaf Bachmann
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)


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


Arithmetic Operation Simplification Rule Computer Algebra System Cost Index Common Lisp 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Wien 1998

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  • Olaf Bachmann

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