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view scripts/ode/private/runge_kutta_45_dorpri.m @ 31706:597f3ee61a48 stable
update Octave Project Developers copyright for the new year
author | John W. Eaton <jwe@octave.org> |
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date | Fri, 06 Jan 2023 13:11:27 -0500 |
parents | e1788b1a315f |
children | 2e484f9f1f18 |
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######################################################################## ## ## Copyright (C) 2013-2023 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## 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 ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}) ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options}) ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals}) ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals}, @var{t_next}) ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_45_dorpri (@dots{}) ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_45_dorpri (@dots{}) ## ## 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 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 an FSAL scheme. ## ## Seventh input parameter is optional and is the time (@var{t_next}) to ## integrate to. The default is @code{@var{t} + @var{dt}}. ## ## 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 an FSAL scheme or for dense output. ## @end deftypefn function [t_next, x_next, x_est, k] = runge_kutta_45_dorpri (fcn, t, x, dt, options = [], k_vals = [], t_next = t + dt) ## Reference: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (2008), ## Solving ordinary differential equations I: Nonstiff problems, ## Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0 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]; 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]; 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 (options)) # extra arguments for function evaluator args = options.funarguments; 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 (fcn, t, x, args{:}); endif k(:,2) = feval (fcn, s(2), x + k(:,1) * aa(2, 1).' , args{:}); k(:,3) = feval (fcn, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:}); k(:,4) = feval (fcn, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:}); k(:,5) = feval (fcn, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:}); k(:,6) = feval (fcn, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:}); ## compute new time and new values for the unknowns ## t_next = t + dt; x_next = 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 (fcn, t_next, x_next, args{:}); cc_prime = dt * c_prime; x_est = x + k * cc_prime(:); endif endfunction