Mercurial > octave-nkf
annotate scripts/ode/private/runge_kutta_45_dorpri.m @ 20584:eb9e2d187ed2
maint: Use Octave coding conventions in scripts/ode/private dir.
* AbsRel_Norm.m, fuzzy_compare.m, hermite_quartic_interpolation.m,
integrate_adaptive.m, integrate_const.m, integrate_n_steps.m, kahan.m,
ode_struct_value_check.m, odepkg_event_handle.m, odepkg_structure_check.m,
runge_kutta_45_dorpri.m, starting_stepsize.m:
Wrap long lines to < 80 chars.
Use double quotes rather than single quotes where possible.
Use ';' at end of keywords "return;" and "break;"
Use '##" for stand-alone comments and '#' for end-of-line comments.
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author | Rik <rik@octave.org> |
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date | Sun, 04 Oct 2015 22:18:54 -0700 |
parents | 25623ef2ff4f |
children | b7ac1e94266e |
rev | line source |
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1 ## Copyright (C) 2013, Roberto Porcu' <roberto.porcu@polimi.it> |
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2 ## Copyright (C) 2015, Carlo de Falco |
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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_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals_in}) |
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22 ## @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}) |
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23 ## @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}) |
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24 ## |
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25 ## This function can be used to integrate a system of ODEs with a given initial |
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26 ## condition @var{x} from @var{t} to @var{t+dt}, with the Dormand-Prince method. |
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27 ## For the definition of this method see |
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28 ## @url{http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method}. |
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29 ## |
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30 ## First output argument is the final integration time value. |
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31 ## |
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32 ## Second output parameter is the higher order computed solution at time |
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33 ## @var{t_next} (local extrapolation). |
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34 ## |
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35 ## Third output parameter is a lower order solution for the estimation of the |
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36 ## error. |
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37 ## |
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38 ## Fourth output parameter is matrix containing the Runge-Kutta evaluations |
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39 ## to use in a FSAL scheme or for dense output. |
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40 ## |
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41 ## First input argument is the function describing the system of ODEs to be |
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42 ## integrated. |
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43 ## |
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44 ## Second input parameter is the first extreme of integration interval. |
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45 ## |
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46 ## Third input argument is the initial condition of the system. |
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47 ## |
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48 ## Fourth input argument is the timestep, that is the length of the |
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49 ## integration interval. |
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50 ## |
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51 ## Fifth input parameter is optional and describes a set of options useful to |
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52 ## adapt the computation to what is needed. |
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53 ## |
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54 ## Sixth input parameter is optional and describes the Runge-Kutta evaluations |
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55 ## of the previous step to use in a FSAL scheme. |
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56 ## @end deftypefn |
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57 ## |
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58 ## @seealso{odepkg} |
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59 |
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60 function [t_out, x_out, x_est, k] = ... |
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61 runge_kutta_45_dorpri (f, t, x, dt, varargin) |
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62 |
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63 persistent a = [0 0 0 0 0 0; |
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64 1/5 0 0 0 0 0; |
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65 3/40 9/40 0 0 0 0; |
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66 44/45 -56/15 32/9 0 0 0; |
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67 19372/6561 -25360/2187 64448/6561 -212/729 0 0; |
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68 9017/3168 -355/33 46732/5247 49/176 -5103/18656 0; |
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69 35/384 0 500/1113 125/192 -2187/6784 11/84]; |
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70 persistent b = [0 1/5 3/10 4/5 8/9 1 1]; |
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71 persistent c = [(35/384) 0 (500/1113) (125/192) (-2187/6784) (11/84)]; |
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72 ## x_est according to Shampine 1986: |
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73 ## persistent c_prime = [(1951/21600) 0 (22642/50085) (451/720), ... |
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74 ## (-12231/42400) (649/6300) (1/60)]; |
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75 persistent c_prime = [(5179/57600) 0 (7571/16695) (393/640), ... |
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76 (-92097/339200) (187/2100) (1/40)]; |
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77 |
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78 s = t + dt * b; |
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79 cc = dt * c; |
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80 aa = dt * a; |
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81 k = zeros (rows (x), 7); |
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82 |
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83 if (nargin >= 5) # options are passed |
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84 args = varargin{1}.vfunarguments; |
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85 if (nargin >= 6) # both the options and the k values are passed |
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86 k(:,1) = varargin{2}(:,end); # FSAL property |
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87 else |
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88 k(:,1) = feval (f, t, x, args{:}); |
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89 endif |
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90 else |
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91 args = {}; |
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92 endif |
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93 |
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94 k(:,2) = feval (f, s(2), x + k(:,1) * aa(2, 1).', args{:}); |
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95 k(:,3) = feval (f, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:}); |
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96 k(:,4) = feval (f, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:}); |
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97 k(:,5) = feval (f, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:}); |
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98 k(:,6) = feval (f, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:}); |
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jcorno <jacopo.corno@gmail.com>
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99 |
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100 ## compute new time and new values for the unknowns |
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Carlo de Falco <carlo.defalco@polimi.it>
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101 t_out = t + dt; |
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102 x_out = x + k(:,1:6) * cc(:); # 5th order approximation |
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103 |
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104 ## if the estimation of the error is required |
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jcorno <jacopo.corno@gmail.com>
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105 if (nargout >= 3) |
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jcorno <jacopo.corno@gmail.com>
parents:
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106 ## new solution to be compared with the previous one |
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Carlo de Falco <carlo.defalco@polimi.it>
parents:
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107 k(:,7) = feval (f, t + dt, x_out, args{:}); |
20572
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Clean up and vetorize Dormant&Prince RK timestepper
Carlo de Falco <carlo.defalco@polimi.it>
parents:
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108 cc_prime = dt * c_prime; |
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Rik <rik@octave.org>
parents:
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109 x_est = x + k * cc_prime(:); |
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parents:
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110 endif |
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111 |
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112 endfunction |
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113 |