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doc: Peridoc grammarcheck of documentation. * container.txi, external.txi, install.txi, oop.txi, plot.txi, testfun.txi, eig.cc, graphics.cc, gsvd.cc, hex2num.cc, pr-output.cc, sub2ind.cc, __ode15__.cc, Map.m, __publish_html_output__.m, __publish_latex_output__.m, odeset.m, ode_event_handler.m, runge_kutta_45_dorpri.m, xlim.m, ylim.m, zlim.m, hgtransform.m, eigs.m, native2unicode.m, unicode2native.m: grammarcheck documentation.
author Rik <rik@octave.org>
date Thu, 08 Jun 2017 17:41:32 -0700
parents 092078913d54
children 194eb4bd202b
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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1 ## Copyright (C) 2016-2017 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
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9 ## (at 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
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14 ## GNU 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 {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt})
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22 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options})
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23 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals})
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24 ## @deftypefnx {} {[@var{t_next}, @var{x_next}] =} runge_kutta_45_dorpri (@var{@@fun}, @var{t}, @var{x}, @var{dt}, @var{options}, @var{k_vals}, @var{t_next})
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25 ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}] =} runge_kutta_45_dorpri (@dots{})
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26 ## @deftypefnx {} {[@var{t_next}, @var{x_next}, @var{x_est}, @var{k_vals_out}] =} runge_kutta_45_dorpri (@dots{})
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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 input argument is the function describing the system of ODEs to be
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34 ## integrated.
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35 ##
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36 ## Second input parameter is the first extreme of integration interval.
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37 ##
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38 ## Third input argument is the initial condition of the system.
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39 ##
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40 ## Fourth input argument is the timestep, that is the length of the
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41 ## integration interval.
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42 ##
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43 ## Fifth input parameter is optional and describes a set of options useful to
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44 ## adapt the computation to what is needed.
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45 ##
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46 ## Sixth input parameter is optional and describes the Runge-Kutta evaluations
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47 ## of the previous step to use in an FSAL scheme.
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48 ##
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49 ## Seventh input parameter is optional and is the time (@var{t_next}) to
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50 ## integrate to. The default is @code{@var{t} + @var{dt}}.
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51 ##
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52 ## First output argument is the final integration time value.
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53 ##
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54 ## Second output parameter is the higher order computed solution at time
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55 ## @var{t_next} (local extrapolation).
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56 ##
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57 ## Third output parameter is a lower order solution for the estimation of the
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58 ## error.
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59 ##
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60 ## Fourth output parameter is matrix containing the Runge-Kutta evaluations
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61 ## to use in an FSAL scheme or for dense output.
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62 ## @end deftypefn
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63
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64 function [t_next, x_next, x_est, k] = runge_kutta_45_dorpri (fun, t, x, dt,
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65 options = [],
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66 k_vals = [],
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67 t_next = t + dt)
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68
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69 ## Reference: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (2008),
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70 ## Solving ordinary differential equations I: Nonstiff problems,
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71 ## Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0
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72 persistent a = [0 0 0 0 0 0;
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73 1/5 0 0 0 0 0;
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74 3/40 9/40 0 0 0 0;
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75 44/45 -56/15 32/9 0 0 0;
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76 19372/6561 -25360/2187 64448/6561 -212/729 0 0;
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77 9017/3168 -355/33 46732/5247 49/176 -5103/18656 0];
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78 persistent b = [0, 1/5, 3/10, 4/5, 8/9, 1, 1];
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79 persistent c = [35/384, 0, 500/1113, 125/192, -2187/6784, 11/84];
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80 persistent c_prime = [5179/57600, 0, 7571/16695, 393/640, ...
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81 -92097/339200, 187/2100, 1/40];
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82
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83 s = t + dt * b;
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84 cc = dt * c;
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85 aa = dt * a;
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86 k = zeros (rows (x), 7);
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87
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88 if (! isempty (options)) # extra arguments for function evaluator
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89 args = options.funarguments;
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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 if (! isempty (k_vals)) # k values from previous step are passed
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95 k(:,1) = k_vals(:,end); # FSAL property
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96 else
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97 k(:,1) = feval (fun, t, x, args{:});
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98 endif
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99
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100 k(:,2) = feval (fun, s(2), x + k(:,1) * aa(2, 1).' , args{:});
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101 k(:,3) = feval (fun, s(3), x + k(:,1:2) * aa(3, 1:2).', args{:});
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102 k(:,4) = feval (fun, s(4), x + k(:,1:3) * aa(4, 1:3).', args{:});
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103 k(:,5) = feval (fun, s(5), x + k(:,1:4) * aa(5, 1:4).', args{:});
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104 k(:,6) = feval (fun, s(6), x + k(:,1:5) * aa(6, 1:5).', args{:});
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105
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106 ## compute new time and new values for the unknowns
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107 ## t_next = t + dt;
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108 x_next = x + k(:,1:6) * cc(:); # 5th order approximation
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109
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110 ## if the estimation of the error is required
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111 if (nargout >= 3)
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112 ## new solution to be compared with the previous one
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113 k(:,7) = feval (fun, t_next, x_next, args{:});
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114 cc_prime = dt * c_prime;
20552
eb9e2d187ed2 maint: Use Octave coding conventions in scripts/ode/private dir.
Rik <rik@octave.org>
parents: 20548
diff changeset
115 x_est = x + k * cc_prime(:);
20533
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
116 endif
20634
80e630b37ba1 maint: Remove unnecessary 'v' prefix before variables in ODE m-files.
Rik <rik@octave.org>
parents: 20621
diff changeset
117
20533
fcb792acab9b Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
jcorno <jacopo.corno@gmail.com>
parents:
diff changeset
118 endfunction