Consider the problem of writing a general subroutine for a Runge-Kutta
scheme [13] for a system of ordinary differential equations. The
convention of writing an -dimensional vector in a bold typeface is adopted.
The general s-stage Runge-Kutta method for the initial-value problem
is defined by
where
and the row-sum conditions
are usually assumed to hold. Furthermore, if
for
,
then
the method is said to be an explicit or classical
Runge-Kutta scheme and the
can be calculated by a straightforward
recursion.
Structural information for a problem, should invariably be exploited in a symbolic system, since the resulting performance gains can be significant [4]. The same techniques should be applied when creating a general template file. A lot of the tedium can be removed from the analysis of a large number of problems and greater throughput is enabled. The next section contains such an example.