Mercurial > octave-nkf
annotate scripts/ode/private/runge_kutta_45_dorpri.m @ 20568:fcb792acab9b
Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
* libinterp/corefcn/levenshtein.cc: move function from odepkg into core
* libinterp/corefcn/module.mk: include levenshtein.cc
* scripts/ode: move ode45, odeset, odeget, and all dependencies
from odepkg into core
* scripts/module.mk: include them
* doc/interpreter/diffeq.txi: add documentation for ode45,
odeset, odeget
* NEWS: announce functions included with this changeset
* scripts/help/__unimplemented__.m: removed new functions
author | jcorno <jacopo.corno@gmail.com> |
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date | Thu, 24 Sep 2015 12:58:46 +0200 |
parents | |
children | 6256f6e366ac |
rev | line source |
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20568
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1 ## Copyright (C) 2013, Roberto Porcu' <roberto.porcu@polimi.it> |
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2 ## OdePkg - A package for solving ordinary differential equations and more |
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3 ## |
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4 ## This program is free software; you can redistribute it and/or modify |
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5 ## it under the terms of the GNU General Public License as published by |
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6 ## the Free Software Foundation; either version 2 of the License, or |
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7 ## (at your option) any later version. |
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8 ## |
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9 ## This program is distributed in the hope that it will be useful, |
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10 ## but WITHOUT ANY WARRANTY; without even the implied warranty of |
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11 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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12 ## GNU General Public License for more details. |
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13 ## |
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14 ## You should have received a copy of the GNU General Public License |
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15 ## along with this program; If not, see <http://www.gnu.org/licenses/>. |
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16 |
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17 ## -*- texinfo -*- |
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18 ## @deftypefn {Command} {[@var{t_next}, @var{x_next}] =} |
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19 ## runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, |
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20 ## [@var{options}, @var{k_vals_in}]) |
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21 ## @deftypefnx {Command} {[@var{t_next}, @var{x_next}, @var{x_est}] =} |
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22 ## runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, |
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23 ## [@var{options}, @var{k_vals_in}]) |
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24 ## @deftypefnx {Command} {[@var{t_next}, @var{x_next}, @var{x_est}, |
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25 ## @var{k_vals_out}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, |
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26 ## @var{dt}, [@var{options}, @var{k_vals_in}]) |
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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 Dormand-Prince method. |
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30 ## For the definition of this method see |
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31 ## @url{http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method}. |
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32 ## |
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33 ## First output argument is the final integration time value. |
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34 ## |
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35 ## Second output parameter is the higher order computed solution at time |
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36 ## @var{t_next} (local extrapolation). |
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37 ## |
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38 ## Third output parameter is a lower order solution for the estimation |
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39 ## of the error. |
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40 ## |
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41 ## Fourth output parameter is matrix containing the Runge-Kutta evaluations |
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42 ## to use in a FSAL scheme or for dense output. |
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43 ## |
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44 ## First input argument is the function describing the system of ODEs |
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45 ## to be integrated. |
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46 ## |
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47 ## Second input parameter is the first extreme of integration interval. |
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48 ## |
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49 ## Third input argument is the initial condition of the system. |
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50 ## |
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51 ## Fourth input argument is the timestep, that is the length of the |
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52 ## integration interval. |
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53 ## |
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54 ## Fifth input parameter is optional and describes a set of options useful |
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55 ## to adapt the computation to what is needed. |
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56 ## |
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57 ## Sixth input parameter is optional and describes the Runge-Kutta evaluations |
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58 ## of the previous step to use in a FSAL scheme. |
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59 ## @end deftypefn |
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60 ## |
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61 ## @seealso{odepkg} |
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62 |
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63 function varargout = runge_kutta_45_dorpri (f, t, x, dt, varargin) |
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64 |
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65 options = varargin{1}; |
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66 k = zeros (size (x, 1), 4); |
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67 |
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68 if (nargin == 5) # only the options are passed |
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69 k(:,1) = feval (f, t , x, options.vfunarguments{:}); |
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70 elseif (nargin == 6) # both the options and the k values are passed |
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71 k(:,1) = varargin{2}(:,end); # FSAL property |
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72 endif |
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73 k(:,1) = feval (f, t, x, options.vfunarguments{:}); |
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74 k(:,2) = feval (f, t + (1/5)*dt, ... |
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75 x + dt * (1/5)*k(:,1), ... |
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76 options.vfunarguments{:}); |
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77 k(:,3) = feval (f, t + (3/10)*dt, ... |
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78 x + dt * ((3/40)*k(:,1) + (9/40)*k(:,2)), ... |
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79 options.vfunarguments{:}); |
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80 k(:,4) = feval (f, t + (4/5)*dt, ... |
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81 x + dt * ((44/45)*k(:,1) - (56/15)*k(:,2) + (32/9)*k(:,3)), ... |
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82 options.vfunarguments{:}); |
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83 k(:,5) = feval (f, t + (8/9)*dt, ... |
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84 x + dt * ((19372/6561)*k(:,1) - (25360/2187)*k(:,2) ... |
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85 + (64448/6561)*k(:,3) - (212/729)*k(:,4)), ... |
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86 options.vfunarguments{:}); |
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87 k(:,6) = feval (f, t + dt, ... |
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88 x + dt * ((9017/3168)*k(:,1) - (355/33)*k(:,2) ... |
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89 + (46732/5247)*k(:,3) + (49/176)*k(:,4) ... |
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90 - (5103/18656)*k(:,5)), ... |
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91 options.vfunarguments{:}); |
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92 |
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93 ## compute new time and new values for the unkwnowns |
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94 varargout{1} = t + dt; |
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95 varargout{2} = x + dt * ((35/384)*k(:,1) + (500/1113)*k(:,3) |
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96 + (125/192)*k(:,4) - (2187/6784)*k(:,5) |
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97 + (11/84)*k(:,6)); # 5th 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(:,7) = feval (f, t + dt, varargout{2}, options.vfunarguments{:}); |
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103 ## x_est according to Shampine 1986: |
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104 ## varargout{3} = x + dt * ((1951/21600)*k(:,1) + (22642/50085)*k(:,3) |
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105 ## + (451/720)*k(:,4) - (12231/42400)*k(:,5) |
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106 ## + (649/6300)*k(:,6) + (1/60)*k(:,7)); |
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107 varargout{3} = x + dt * ((5179/57600)*k(:,1) + (7571/16695)*k(:,3) |
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108 + (393/640)*k(:,4) - (92097/339200)*k(:,5) |
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109 + (187/2100)*k(:,6) + (1/40)*k(:,7)); # x_est |
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110 varargout{4} = k; |
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111 endif |
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112 |
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113 endfunction |
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114 |
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115 ## Local Variables: *** |
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116 ## mode: octave *** |
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117 ## End: *** |