* Implement other stepping methods from well-known packages such as
RKSUITE, DASSL, etc

* The entry below has been downgraded from a bug.  We use the
coefficients given in the original paper by Prince and Dormand, and it
is true that these are inexact (the values in the paper are said to be
accurate 18 figures).  If somebody publishes exact versions we will
use them, but at present it is better to stick with the published
versions of the coefficients them use our own.
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BUG#8 -- inexact coefficients in rk8pd.c

From: Luc Maisonobe <Luc.Maisonobe@c-s.fr>
To: gsl-discuss@sources.redhat.com
Subject: further thoughts about Dormand-Prince 8 (RK8PD)
Date: Wed, 14 Aug 2002 10:50:49 +0200

I was looking for some references concerning Runge-Kutta methods when I
noticed GSL had an high order one. I also found a question in the list
archive (April 2002) about the references of this method which is
implemented in rk8pd.c. It was said the coefficients were taken from the
"Numerical Algorithms with C" book by Engeln-Mullges and Uhlig.

I have checked the coefficients somewhat with a little java tool I have
developped (see http://www.spaceroots.org/archive.htm#RKCheckSoftware)
and found they were not exact. I think this method is really the method
that is already in rksuite (http://www.netlib.org/ode/rksuite/) were the
coefficients are given as real values with 30 decimal digits. The
coefficients have probably been approximated as fractions later on.
However, these approximations are not perfect, they are good only for
the first 16 or 18 digits depending on the coefficient.

This has no consequence for practical purposes since they are stored in
double variables, but give a false impression of beeing exact
expressions. Well, there are even some coefficients that should really
be rational numbers but for which wrong numerators and denominators are
given. As an example, the first and fourth elements of the b7 array are
given as 29443841.0 / 614563906.0 and 77736538.0 / 692538347, hence the
sum off all elements of the b7 array (which should theoretically be
equal to ah[5]) only approximate this. For these two coefficients, this
could have been avoided using  215595617.0 / 4500000000.0 and
202047683.0 / 1800000000.0, which also looks more coherent with the
other coefficients.

The rksuite comments say this method is described in this paper :

   High Order Embedded Runge-Kutta Formulae
   P.J. Prince and J.R. Dormand
   J. Comp. Appl. Math.,7, pp. 67-75, 1981

It also says the method is an 8(7) method (i.e. the coefficients set
used to advance integration is order 8 and error estimation is order 7).
If I use my tool to check the order, I am forced to check the order
conditions numerically with a tolerance since I do not have an exact
expression of the coefficients. Since even if some conditions are not
mathematically met, the residuals are small and could be below the
tolerance. There are tolerance values for which such numerical test
dedeuce the method is of order 9, as is said in GSL. However, I am not
convinced, there are to few parameters for the large number of order
conditions needed at order 9.

I would suggest to correct the coefficients in rk8pd.c (just put the
literal constants of rksuite) and to add the reference to the article.

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