[ODE] Speeding up iterative techniques

[Henri Hakl]

I've been wondering about the following for quite a while now - since I
never saw this mentioned anywhere I figured there is either a flaw, or more
likely nobody has considered this possibility:

An iterative approach gives a stab at the correct solution and with each
iteration gets a bit closer to the real solution. The convergence may be
linear or something else. My guess is that the iterative approach discussed
in the ODE mailing list converges linearly (but I don't think it matters all
that much).

Given (any) linearly convergent series, this series can be accelerated using
Aitken's delta-squared method, this method is improved upon with a method
called Steffensen's method - these can make a linearly convergent series
converge quadratically. And the computations involved are >extremely< simple
and fast.

I think it is something promising to look into, if you cannot find
references to the technique let me know, I can forward some more
information.

(If the technique cannot be applied to the iterative approach used currently
(because it is of a higher order then linear), maybe a FAST linear iteration
can be made that is then accelerated to quadratic using Steffensen's
method.)