🚀 go-pugleaf

In article <C1H5uF.5HC@rice.edu> dharry@cyclone.rice.edu (Dennis Harry) writes:
>In article <1993Jan20.054301.455@seas.smu.edu>, mhosea@seas.smu.edu (Mike Hosea) writes:
> 
>|> The reason extrapolation methods get very accurate results with large step
>|> sizes is because they can use very high order approximations.  However,
>|> my research has shown that when compared to conventional Runge-Kutta
>|> and Adams methods of the same order, the extrapolation formulas are not
>|> as efficient in terms of function evaluations per unit step.
>                                                 ^^^^^^^^^^^^^
>Yes, but the point is that they may take much larger steps.  Ideally,
>their relative inefficiency per unit step is (more than) offset by having
>to take fewer steps?

No, it's not.  Not when you consider the competition.  It is true that
an extrapolation code may perform better than a fixed order Runge-Kutta
code when the tolerance is very stringent, but this is only because they
can go to higher order.  However, like the extrapolation formulas, the
Adams-Moulton formulas are variable order, and at each order (especially
high orders) the Adams-Moulton formulas are much more efficient.  If you
compare the performances of codes you may not see this since the Adams
code might not attept to use the higher orders--this would have nothing
to do with the methods themselves.  Anyway, I am not arguing from intuition.
Check out the mathematics in my report or any reputable numerical
comparison like

  L.F. Shampine and L.S. Baca  1986  "Fixed versus variable order
  Runge-Kutta", ACM TOMS, 12, no.  pp. 1-23.
-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)