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maint: Use "fcn" as preferred abbreviation for "function" in m-files. * accumarray.m, accumdim.m, quadl.m, quadv.m, randi.m, structfun.m, __is_function__.m, uigetfile.m, uimenu.m, uiputfile.m, doc_cache_create.m, colorspace_conversion_input_check.m, imageIO.m, argnames.m, vectorize.m, vectorize.m, normest1.m, inputname.m, nthargout.m, display_info_file.m, decic.m, ode15i.m, ode15s.m, ode23.m, ode23s.m, ode45.m, odeset.m, check_default_input.m, integrate_adaptive.m, ode_event_handler.m, runge_kutta_23.m, runge_kutta_23s.m, runge_kutta_45_dorpri.m, runge_kutta_interpolate.m, starting_stepsize.m, __all_opts__.m, fminbnd.m, fminsearch.m, fminunc.m, fsolve.m, fzero.m, sqp.m, fplot.m, plotyy.m, __bar__.m, __ezplot__.m, flat_entry.html, profexport.m, movfun.m, bicg.m, bicgstab.m, cgs.m, eigs.m, gmres.m, pcg.m, __alltohandles__.m, __sprand__.m, qmr.m, tfqmr.m, dump_demos.m: Replace "func", "fun", "fn" in documentation and variable names with "fcn".
author Rik <rik@octave.org>
date Mon, 04 Apr 2022 18:14:56 -0700
parents 796f54d4ddbf
children 597f3ee61a48
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2013-2022 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt})
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28 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options})
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29 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals})
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30 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fcn}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals}, @var{t_next})
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31 ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_45_dorpri (@dots{})
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32 ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_45_dorpri (@dots{})
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33 ##
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34 ## This function can be used to integrate a system of ODEs with a given initial
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35 ## condition @var{x} from @var{t} to @var{t+dt} with the Dormand-Prince method.
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36 ## For the definition of this method see
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37 ## @url{http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method}.
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38 ##
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39 ## First input argument is the function describing the system of ODEs to be
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40 ## integrated.
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41 ##
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42 ## Second input parameter is the first extreme of integration interval.
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43 ##
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44 ## Third input argument is the initial condition of the system.
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45 ##
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46 ## Fourth input argument is the timestep, that is the length of the
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47 ## integration interval.
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48 ##
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49 ## Fifth input parameter is optional and describes a set of options useful to
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50 ## adapt the computation to what is needed.
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51 ##
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52 ## Sixth input parameter is optional and describes the Runge-Kutta evaluations
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53 ## of the previous step to use in an FSAL scheme.
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54 ##
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55 ## Seventh input parameter is optional and is the time (@var{t_next}) to
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56 ## integrate to. The default is @code{@var{t} + @var{dt}}.
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57 ##
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58 ## First output argument is the final integration time value.
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59 ##
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60 ## Second output parameter is the higher order computed solution at time
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61 ## @var{t_next} (local extrapolation).
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62 ##
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63 ## Third output parameter is a lower order solution for the estimation of the
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64 ## error.
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65 ##
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66 ## Fourth output parameter is matrix containing the Runge-Kutta evaluations
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67 ## to use in an FSAL scheme or for dense output.
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68 ## @end deftypefn
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69
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70 function [t_next, x_next, x_est, k] = runge_kutta_45_dorpri (fcn, t, x, dt,
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71 options = [],
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72 k_vals = [],
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73 t_next = t + dt)
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74
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75 ## Reference: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (2008),
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76 ## Solving ordinary differential equations I: Nonstiff problems,
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77 ## Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0
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78 persistent a = [0 0 0 0 0 0;
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79 1/5 0 0 0 0 0;
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80 3/40 9/40 0 0 0 0;
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81 44/45 -56/15 32/9 0 0 0;
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82 19372/6561 -25360/2187 64448/6561 -212/729 0 0;
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83 9017/3168 -355/33 46732/5247 49/176 -5103/18656 0];
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84 persistent b = [0, 1/5, 3/10, 4/5, 8/9, 1, 1];
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85 persistent c = [35/384, 0, 500/1113, 125/192, -2187/6784, 11/84];
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86 persistent c_prime = [5179/57600, 0, 7571/16695, 393/640, ...
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87 -92097/339200, 187/2100, 1/40];
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88
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89 s = t + dt * b;
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90 cc = dt * c;
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91 aa = dt * a;
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92 k = zeros (rows (x), 7);
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93
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94 if (! isempty (options)) # extra arguments for function evaluator
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95 args = options.funarguments;
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96 else
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97 args = {};
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98 endif
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99
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100 if (! isempty (k_vals)) # k values from previous step are passed
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101 k(:,1) = k_vals(:,end); # FSAL property
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102 else
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103 k(:,1) = feval (fcn, t, x, args{:});
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104 endif
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105
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106 k(:,2) = feval (fcn, s(2), x + k(:,1) * aa(2, 1).' , args{:});
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107 k(:,3) = feval (fcn, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:});
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108 k(:,4) = feval (fcn, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:});
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109 k(:,5) = feval (fcn, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:});
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110 k(:,6) = feval (fcn, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:});
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111
20552
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Rik <rik@octave.org>
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112 ## compute new time and new values for the unknowns
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Rik <rik@octave.org>
parents: 20600
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113 ## t_next = t + dt;
b92f8e148936 maint: Continued clean-up of functions in ode/private dir.
Rik <rik@octave.org>
parents: 20600
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114 x_next = x + k(:,1:6) * cc(:); # 5th order approximation
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parents:
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115
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jcorno <jacopo.corno@gmail.com>
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116 ## if the estimation of the error is required
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117 if (nargout >= 3)
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118 ## new solution to be compared with the previous one
30893
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Rik <rik@octave.org>
parents: 30564
diff changeset
119 k(:,7) = feval (fcn, t_next, x_next, args{:});
20540
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Carlo de Falco <carlo.defalco@polimi.it>
parents: 20536
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120 cc_prime = dt * c_prime;
20552
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Rik <rik@octave.org>
parents: 20548
diff changeset
121 x_est = x + k * cc_prime(:);
20533
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jcorno <jacopo.corno@gmail.com>
parents:
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122 endif
20634
80e630b37ba1 maint: Remove unnecessary 'v' prefix before variables in ODE m-files.
Rik <rik@octave.org>
parents: 20621
diff changeset
123
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jcorno <jacopo.corno@gmail.com>
parents:
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124 endfunction