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Fix copyright text in private ode functions * script/ode/private/*.m: Fox the copyright text.
author Carlo de Falco <carlo.defalco@polimi.it>
date Fri, 02 Oct 2015 05:50:43 +0200
parents fcb792acab9b
children 72cd24aa5f7a
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1 ## Copyright (C) 2013, Roberto Porcu' <roberto.porcu@polimi.it>
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Command} {[@var{t_next}, @var{x_next}] =}
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21 ## runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt},
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22 ## [@var{options}, @var{k_vals_in}])
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23 ## @deftypefnx {Command} {[@var{t_next}, @var{x_next}, @var{x_est}] =}
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24 ## runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt},
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25 ## [@var{options}, @var{k_vals_in}])
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26 ## @deftypefnx {Command} {[@var{t_next}, @var{x_next}, @var{x_est},
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27 ## @var{k_vals_out}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x},
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28 ## @var{dt}, [@var{options}, @var{k_vals_in}])
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29 ##
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30 ## This function can be used to integrate a system of ODEs with a given initial
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31 ## condition @var{x} from @var{t} to @var{t+dt}, with the Dormand-Prince method.
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32 ## For the definition of this method see
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33 ## @url{http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method}.
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34 ##
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35 ## First output argument is the final integration time value.
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36 ##
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37 ## Second output parameter is the higher order computed solution at time
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38 ## @var{t_next} (local extrapolation).
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39 ##
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40 ## Third output parameter is a lower order solution for the estimation
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41 ## of the error.
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42 ##
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43 ## Fourth output parameter is matrix containing the Runge-Kutta evaluations
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44 ## to use in a FSAL scheme or for dense output.
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45 ##
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46 ## First input argument is the function describing the system of ODEs
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47 ## to be integrated.
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48 ##
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49 ## Second input parameter is the first extreme of integration interval.
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50 ##
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51 ## Third input argument is the initial condition of the system.
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52 ##
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53 ## Fourth input argument is the timestep, that is the length of the
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54 ## integration interval.
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55 ##
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56 ## Fifth input parameter is optional and describes a set of options useful
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57 ## to adapt the computation to what is needed.
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58 ##
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59 ## Sixth input parameter is optional and describes the Runge-Kutta evaluations
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60 ## of the previous step to use in a FSAL scheme.
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61 ## @end deftypefn
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62 ##
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63 ## @seealso{odepkg}
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64
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65 function varargout = runge_kutta_45_dorpri (f, t, x, dt, varargin)
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66
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67 options = varargin{1};
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68 k = zeros (rows (x), 6);
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69
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70 if (nargin == 5) # only the options are passed
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71 k(:,1) = feval (f, t , x, options.vfunarguments{:});
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72 elseif (nargin == 6) # both the options and the k values are passed
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73 k(:,1) = varargin{2}(:,end); # FSAL property
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74 endif
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75
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76 k(:,1) = feval (f, t, x, options.vfunarguments{:});
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77 k(:,2) = feval (f, t + (1/5)*dt, ...
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78 x + dt * (1/5)*k(:,1), ...
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79 options.vfunarguments{:});
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80 k(:,3) = feval (f, t + (3/10)*dt, ...
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81 x + dt * ((3/40)*k(:,1) + (9/40)*k(:,2)), ...
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82 options.vfunarguments{:});
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83 k(:,4) = feval (f, t + (4/5)*dt, ...
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84 x + dt * ((44/45)*k(:,1) - (56/15)*k(:,2) + (32/9)*k(:,3)), ...
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85 options.vfunarguments{:});
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86 k(:,5) = feval (f, t + (8/9)*dt, ...
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87 x + dt * ((19372/6561)*k(:,1) - (25360/2187)*k(:,2) ...
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88 + (64448/6561)*k(:,3) - (212/729)*k(:,4)), ...
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89 options.vfunarguments{:});
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90 k(:,6) = feval (f, t + dt, ...
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91 x + dt * ((9017/3168)*k(:,1) - (355/33)*k(:,2) ...
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92 + (46732/5247)*k(:,3) + (49/176)*k(:,4) ...
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93 - (5103/18656)*k(:,5)), ...
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94 options.vfunarguments{:});
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95
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96 ## compute new time and new values for the unkwnowns
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97 varargout{1} = t + dt;
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98 varargout{2} = x + dt * ((35/384)*k(:,1) + (500/1113)*k(:,3)
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99 + (125/192)*k(:,4) - (2187/6784)*k(:,5)
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100 + (11/84)*k(:,6)); # 5th order approximation
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101
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102 ## if the estimation of the error is required
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103 if (nargout >= 3)
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104 ## new solution to be compared with the previous one
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105 k(:,7) = feval (f, t + dt, varargout{2}, options.vfunarguments{:});
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106 ## x_est according to Shampine 1986:
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107 ## varargout{3} = x + dt * ((1951/21600)*k(:,1) + (22642/50085)*k(:,3)
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108 ## + (451/720)*k(:,4) - (12231/42400)*k(:,5)
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109 ## + (649/6300)*k(:,6) + (1/60)*k(:,7));
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
110 varargout{3} = x + dt * ((5179/57600)*k(:,1) + (7571/16695)*k(:,3)
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
111 + (393/640)*k(:,4) - (92097/339200)*k(:,5)
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
112 + (187/2100)*k(:,6) + (1/40)*k(:,7)); # x_est
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
113 varargout{4} = k;
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
114 endif
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
115
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
116 endfunction
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
117