Computer Algebra Recipes

2.3. NUMERICAL SOLUTION OF ODES 93
round-o® error mentioned earlier. To compare the speed of di®erent numerical
schemes or algorithms, the computer CPU time at which the execution of the
algorithm begins is recorded using the time command. The CPU time at which
the execution is ¯nished will also be recorded, the CPU time for the algorithm
then being the di®erence. The CPU time will depend on the computer being
used, the times quoted here being for a 3-GHz PC.
> Digits:=10: begin:=time():
Functional operators are formed to calculate R and F for speci¯ed x and y.
> R:=(x,y)->a*x-b*x*y: F:=(x,y)->-c*y+d*x*y:
A do loop is used to iterate the di®erence equations (2.16) and the time.
> for n from 0 to N do
> r[n+1]:=r[n]+h*R(r[n],f[n]);
> f[n+1]:=f[n]+h*F(r[n],f[n]);
> t[n+1]:=t[n]+h;
A 3-dimensional plotting point is formed for the nth time step.
> pnt[n]:=[t[n],r[n],f[n]];
> end do:
On ending the loop, the temporal evolution of the solution will be observed by
creating a three-dimensional plot using a sequence of the plotting points from
n =0ton = N and the pointplot3d command.
> pointplot3d([seq(pnt[n],n=0..N)],symbol=cross,color=red,
axes=normal,labels=["t","r","f"],tickmarks=[2,3,3]);
20
t
600
f
400
200
0 0
200
r
400
Figure 2.22: Forward Euler solution of the rabbits{foxes equations.