Mercurial > octave-nkf
view scripts/ode/private/runge_kutta_45_dorpri.m @ 20654:b65888ec820e draft default tip gccjit
dmalcom gcc jit import
author | Stefan Mahr <dac922@gmx.de> |
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date | Fri, 27 Feb 2015 16:59:36 +0100 |
parents | a22d8a2eb0e5 |
children |
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## Copyright (C) 2015, Carlo de Falco ## Copyright (C) 2013, Roberto Porcu' <roberto.porcu@polimi.it> ## ## This file is part of Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in}) ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in}) ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in}) ## ## This function can be used to integrate a system of ODEs with a given initial ## condition @var{x} from @var{t} to @var{t+dt}, with the Dormand-Prince method. ## For the definition of this method see ## @url{http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method}. ## ## First output argument is the final integration time value. ## ## Second output parameter is the higher order computed solution at time ## @var{t_next} (local extrapolation). ## ## Third output parameter is a lower order solution for the estimation of the ## error. ## ## Fourth output parameter is matrix containing the Runge-Kutta evaluations ## to use in a FSAL scheme or for dense output. ## ## First input argument is the function describing the system of ODEs to be ## integrated. ## ## Second input parameter is the first extreme of integration interval. ## ## Third input argument is the initial condition of the system. ## ## Fourth input argument is the timestep, that is the length of the ## integration interval. ## ## Fifth input parameter is optional and describes a set of options useful to ## adapt the computation to what is needed. ## ## Sixth input parameter is optional and describes the Runge-Kutta evaluations ## of the previous step to use in a FSAL scheme. ## @end deftypefn ## ## @seealso{odepkg} function [t_out, x_out, x_est, k] = ... runge_kutta_45_dorpri (f, t, x, dt, opts = [], k_vals = [], t_out = t + dt) persistent a = [0 0 0 0 0 0; 1/5 0 0 0 0 0; 3/40 9/40 0 0 0 0; 44/45 -56/15 32/9 0 0 0; 19372/6561 -25360/2187 64448/6561 -212/729 0 0; 9017/3168 -355/33 46732/5247 49/176 -5103/18656 0; 35/384 0 500/1113 125/192 -2187/6784 11/84]; persistent b = [0 1/5 3/10 4/5 8/9 1 1]; persistent c = [(35/384) 0 (500/1113) (125/192) (-2187/6784) (11/84)]; ## x_est according to Shampine 1986: ## persistent c_prime = [(1951/21600) 0 (22642/50085) (451/720), ... ## (-12231/42400) (649/6300) (1/60)]; persistent c_prime = [(5179/57600) 0 (7571/16695) (393/640), ... (-92097/339200) (187/2100) (1/40)]; s = t + dt * b; cc = dt * c; aa = dt * a; k = zeros (rows (x), 7); if (! isempty (opts)) # extra arguments for function evaluator args = opts.vfunarguments; else args = {}; endif if (! isempty (k_vals)) # k values from previous step are passed k(:,1) = k_vals(:,end); # FSAL property else k(:,1) = feval (f, t, x, args{:}); endif k(:,2) = feval (f, s(2), x + k(:,1) * aa(2, 1).', args{:}); k(:,3) = feval (f, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:}); k(:,4) = feval (f, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:}); k(:,5) = feval (f, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:}); k(:,6) = feval (f, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:}); ## compute new time and new values for the unknowns ## t_out = t + dt; x_out = x + k(:,1:6) * cc(:); # 5th order approximation ## if the estimation of the error is required if (nargout >= 3) ## new solution to be compared with the previous one k(:,7) = feval (f, t_out, x_out, args{:}); cc_prime = dt * c_prime; x_est = x + k * cc_prime(:); endif endfunction