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annotate scripts/ode/private/runge_kutta_23.m @ 20903:3d3da166dac5

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2015 Code Sprint: finish import of ode23 into core * scripts/ode/private/runge_kutta_23.m: apply vectorization. * scripts/ode/private/runge_kutta_45_dorpri.m: remove unused parts of the tableau. * scripts/ode/private/runge_kutta_interpolate.m: reimplement cubic hermite interpolation.
author Carlo de Falco <carlo.defalco@polimi.it>
date Tue, 15 Dec 2015 18:50:58 +0100
parents afe9c529760d
children 2b8447888e0a
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20901
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1 ## Copyright (C) 2013-2015, Jacopo Corno <jacopo.corno@gmail.com>
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2 ## Copyright (C) 2013-2015, Roberto Porcu' <roberto.porcu@polimi.it>
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {Function File} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{@fun}, @var{t}, @var{x}, @var{dt})
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22 ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{@fun}, @var{t}, @var{x}, @var{dt}, @var{options})
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23 ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in})
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24 ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}] =} runge_kutta_23 (@var{@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in}, @var{t_next})
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25 ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_23 (@dots{})
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26 ## @deftypefnx {Function File} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_23 (@dots{})
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27 ##
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28 ## This function can be used to integrate a system of ODEs with a given initial
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29 ## condition @var{x} from @var{t} to @var{t+dt}, with the Bogacki-Shampine
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30 ## method of third order. For the definition of this method see
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31 ## @url{http://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods}.
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32 ##
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33 ## @var{@fun} is a function handle that defines the ODE: @code{y' = f(tau,y)}.
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34 ## The function must accept two inputs where the first is time @var{tau} and the
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35 ## second is a column vector of unknowns @var{y}.
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36 ##
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37 ## @var{t} is the first extreme of integration interval.
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38 ##
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39 ## @var{x} is the initial condition of the system..
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40 ##
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41 ## @var{dt} is the timestep, that is the length of the integration interval.
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42 ##
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43 ## The optional fourth argument @var{options} specifies options for the ODE
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44 ## solver. It is a structure generated by @code{odeset}. In particular it
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45 ## contains the field @var{funarguments} with the optional arguments to be used
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46 ## in the evaluation of @var{fun}.
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47 ##
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48 ## The optional fifth argument @var{k_vals_in} contains the Runge-Kutta
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49 ## evaluations of the previous step to use in a FSAL scheme.
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50 ##
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51 ## The optional sixth argument @var{t_next} (@code{t_next = t + dt}) specifies
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52 ## the end of the integration interval. The output @var{x_next} s the higher
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53 ## order computed solution at time @var{t_next} (local extrapolation is
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54 ## performed).
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55 ##
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56 ## Optionally the functions can also return @var{x_est}, a lower order solution
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57 ## for the estimation of the error, and @var{k_vals_out}, a matrix containing
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58 ## the Runge-Kutta evaluations to use in a FSAL scheme or for dense output.
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59 ##
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60 ## @seealso{runge_kutta_45_dorpri}
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61 ## @end deftypefn
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62
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63 function [t_next, x_next, x_est, k] = runge_kutta_23 (f, t, x, dt,
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64 options = [],
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65 k_vals = [],
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66 t_next = t + dt)
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67
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68 persistent a = [0 0 0;
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69 1/2 0 0;
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70 0 3/4 0];
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71 persistent b = [0 1/2 3/4 1];
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72 persistent c = [(2/9) (1/3) (4/9)];
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73 persistent c_prime = [(7/24) (1/4) (1/3) (1/8)];
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74
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75 s = t + dt * b;
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76 cc = dt * c;
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77 aa = dt * a;
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78 k = zeros (rows (x), 4);
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79
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80 if (! isempty (options)) # extra arguments for function evaluator
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81 args = options.funarguments;
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82 else
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83 args = {};
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84 endif
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85
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86 if (! isempty (k_vals)) # k values from previous step are passed
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87 k(:,1) = k_vals(:,end); # FSAL property
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88 else
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89 k(:,1) = feval (f, t, x, args{:});
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90 endif
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91
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92 k(:,2) = feval (f, s(2), x + k(:,1) * aa(2, 1).', args{:});
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93 k(:,3) = feval (f, s(3), x + k(:,2) * aa(3, 2).', args{:});
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94
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95 ## compute new time and new values for the unkwnowns
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96 ## t_next = t + dt;
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97 x_next = x + k(:,1:3) * cc(:); # 3rd order approximation
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98
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99 ## if the estimation of the error is required
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100 if (nargout >= 3)
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101 ## new solution to be compared with the previous one
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102 k(:,4) = feval (f, t_next, x_next, args{:});
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103 cc_prime = dt * c_prime;
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104 x_est = x + k * cc_prime(:);
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afe9c529760d 2015 Code Sprint: move ode23 and runge_kutta_23 from odepkg to core
Stefan Miereis <stefan.miereis@gmx.de>
parents:
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105 endif
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106
afe9c529760d 2015 Code Sprint: move ode23 and runge_kutta_23 from odepkg to core
Stefan Miereis <stefan.miereis@gmx.de>
parents:
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107 endfunction