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annotate scripts/ode/private/runge_kutta_45_dorpri.m @ 20654:b65888ec820e draft default tip gccjit

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dmalcom gcc jit import
author Stefan Mahr <dac922@gmx.de>
date Fri, 27 Feb 2015 16:59:36 +0100
parents a22d8a2eb0e5
children
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rev   line source
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1 ## Copyright (C) 2015, Carlo de Falco
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2 ## Copyright (C) 2013, 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_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, opts = [], k_vals = [],
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62 t_out = t + dt)
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63
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64 persistent a = [0 0 0 0 0 0;
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65 1/5 0 0 0 0 0;
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66 3/40 9/40 0 0 0 0;
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67 44/45 -56/15 32/9 0 0 0;
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68 19372/6561 -25360/2187 64448/6561 -212/729 0 0;
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69 9017/3168 -355/33 46732/5247 49/176 -5103/18656 0;
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70 35/384 0 500/1113 125/192 -2187/6784 11/84];
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71 persistent b = [0 1/5 3/10 4/5 8/9 1 1];
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72 persistent c = [(35/384) 0 (500/1113) (125/192) (-2187/6784) (11/84)];
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73 ## x_est according to Shampine 1986:
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74 ## persistent c_prime = [(1951/21600) 0 (22642/50085) (451/720), ...
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75 ## (-12231/42400) (649/6300) (1/60)];
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76 persistent c_prime = [(5179/57600) 0 (7571/16695) (393/640), ...
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77 (-92097/339200) (187/2100) (1/40)];
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78
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79 s = t + dt * b;
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80 cc = dt * c;
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81 aa = dt * a;
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82 k = zeros (rows (x), 7);
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83
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84 if (! isempty (opts)) # extra arguments for function evaluator
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85 args = opts.vfunarguments;
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86 else
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87 args = {};
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88 endif
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89
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90 if (! isempty (k_vals)) # k values from previous step are passed
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91 k(:,1) = k_vals(:,end); # FSAL property
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92 else
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93 k(:,1) = feval (f, t, x, args{:});
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94 endif
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95
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96 k(:,2) = feval (f, s(2), x + k(:,1) * aa(2, 1).', args{:});
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97 k(:,3) = feval (f, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:});
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98 k(:,4) = feval (f, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:});
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99 k(:,5) = feval (f, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:});
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100 k(:,6) = feval (f, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:});
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101
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102 ## compute new time and new values for the unknowns
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103 ## t_out = t + dt;
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104 x_out = x + k(:,1:6) * cc(:); # 5th order approximation
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105
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106 ## if the estimation of the error is required
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107 if (nargout >= 3)
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108 ## new solution to be compared with the previous one
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109 k(:,7) = feval (f, t_out, x_out, args{:});
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110 cc_prime = dt * c_prime;
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111 x_est = x + k * cc_prime(:);
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fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
112 endif
20572
72cd24aa5f7a Clean up and vetorize Dormant&Prince RK timestepper
Carlo de Falco <carlo.defalco@polimi.it>
parents: 20570
diff changeset
113
20568
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
114 endfunction
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
115