# -*- org -*- #+CATEGORY: ode-initval * Add a higher level interface which accepts a start point, end point, result array (size N, y0, y1, y2 ... ,y(N-1)) desired relative and absolute errors epsrel and epsabs it should have its own workspace which is a wrapper around the existing workspaces * Implement other stepping methods from well-known packages such as RKSUITE, VODE, DASSL, etc * Roundoff error needs to be taken into account to prevent the step-size being made arbitrarily small * 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. ---------------------------------------------------------------------- BUG#8 -- inexact coefficients in rk8pd.c From: Luc Maisonobe 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. ----------------------------------------------------------------------