Mercurial > octave
annotate scripts/ode/ode45.m @ 23219:3ac9f9ecfae5 stable
maint: Update copyright dates.
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 22 Feb 2017 12:39:29 -0500 |
parents | e9a0469dedd9 |
children | 092078913d54 |
rev | line source |
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1 ## Copyright (C) 2016-2017 Carlo de Falco |
22626 | 2 ## Copyright (C) 2016 Francesco Faccio <francesco.faccio@mail.polimi.it> |
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3 ## Copyright (C) 2014-2016 Jacopo Corno <jacopo.corno@gmail.com> |
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4 ## Copyright (C) 2013-2016 Roberto Porcu' <roberto.porcu@polimi.it> |
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5 ## Copyright (C) 2006-2012 Thomas Treichl <treichl@users.sourceforge.net> |
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6 ## |
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7 ## This file is part of Octave. |
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8 ## |
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9 ## Octave is free software; you can redistribute it and/or modify it |
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10 ## under the terms of the GNU General Public License as published by |
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11 ## the Free Software Foundation; either version 3 of the License, or (at |
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12 ## your option) any later version. |
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13 ## |
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14 ## Octave is distributed in the hope that it will be useful, but |
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15 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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16 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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17 ## General Public License for more details. |
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18 ## |
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19 ## You should have received a copy of the GNU General Public License |
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20 ## along with Octave; see the file COPYING. If not, see |
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21 ## <http://www.gnu.org/licenses/>. |
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22 |
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23 ## -*- texinfo -*- |
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24 ## @deftypefn {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}) |
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25 ## @deftypefnx {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}, @var{ode_opt}) |
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26 ## @deftypefnx {} {[@var{t}, @var{y}, @var{te}, @var{ye}, @var{ie}] =} ode45 (@dots{}) |
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27 ## @deftypefnx {} {@var{solution} =} ode45 (@dots{}) |
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28 ## @deftypefnx {} {} ode45 (@dots{}) |
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29 ## |
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30 ## Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) |
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31 ## with the well known explicit @nospell{Dormand-Prince} method of order 4. |
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32 ## |
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33 ## @var{fun} is a function handle, inline function, or string containing the |
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34 ## name of the function that defines the ODE: @code{y' = f(t,y)}. The function |
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35 ## must accept two inputs where the first is time @var{t} and the second is a |
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36 ## column vector of unknowns @var{y}. |
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37 ## |
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38 ## @var{trange} specifies the time interval over which the ODE will be |
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39 ## evaluated. Typically, it is a two-element vector specifying the initial and |
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40 ## final times (@code{[tinit, tfinal]}). If there are more than two elements |
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41 ## then the solution will also be evaluated at these intermediate time |
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42 ## instances. |
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43 ## |
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44 ## By default, @code{ode45} uses an adaptive timestep with the |
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45 ## @code{integrate_adaptive} algorithm. The tolerance for the timestep |
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46 ## computation may be changed by using the options @qcode{"RelTol"} |
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47 ## and @qcode{"AbsTol"}. |
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48 ## |
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49 ## @var{init} contains the initial value for the unknowns. If it is a row |
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50 ## vector then the solution @var{y} will be a matrix in which each column is |
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51 ## the solution for the corresponding initial value in @var{init}. |
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52 ## |
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53 ## The optional fourth argument @var{ode_opt} specifies non-default options to |
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54 ## the ODE solver. It is a structure generated by @code{odeset}. |
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55 ## |
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56 ## The function typically returns two outputs. Variable @var{t} is a |
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57 ## column vector and contains the times where the solution was found. The |
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58 ## output @var{y} is a matrix in which each column refers to a different |
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59 ## unknown of the problem and each row corresponds to a time in @var{t}. |
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60 ## |
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61 ## The output can also be returned as a structure @var{solution} which |
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62 ## has a field @var{x} containing a row vector of times where the solution |
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63 ## was evaluated and a field @var{y} containing the solution matrix such |
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64 ## that each column corresponds to a time in @var{x}. |
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65 ## Use @code{fieldnames (@var{solution})} to see the other fields and |
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66 ## additional information returned. |
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67 ## |
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68 ## If no output arguments are requested, and no @code{OutputFcn} is |
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69 ## specified in @var{ode_opt}, then the @code{OutputFcn} is set to |
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70 ## @code{odeplot} and the results of the solver are plotted immediately. |
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71 ## |
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72 ## If using the @qcode{"Events"} option then three additional outputs may |
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73 ## be returned. @var{te} holds the time when an Event function returned a |
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74 ## zero. @var{ye} holds the value of the solution at time @var{te}. @var{ie} |
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75 ## contains an index indicating which Event function was triggered in the case |
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76 ## of multiple Event functions. |
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77 ## |
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78 ## Example: Solve the @nospell{Van der Pol} equation |
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79 ## |
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80 ## @example |
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81 ## @group |
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82 ## fvdp = @@(@var{t},@var{y}) [@var{y}(2); (1 - @var{y}(1)^2) * @var{y}(2) - @var{y}(1)]; |
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83 ## [@var{t},@var{y}] = ode45 (fvdp, [0, 20], [2, 0]); |
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84 ## @end group |
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85 ## @end example |
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87 ## @end deftypefn |
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88 |
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89 function varargout = ode45 (fun, trange, init, varargin) |
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90 |
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91 if (nargin < 3) |
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92 print_usage (); |
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93 endif |
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94 |
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95 order = 5; # runge_kutta_45_dorpri uses local extrapolation |
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96 solver = "ode45"; |
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97 |
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98 if (nargin >= 4) |
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99 if (! isstruct (varargin{1})) |
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100 ## varargin{1:len} are parameters for fun |
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101 odeopts = odeset (); |
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102 funarguments = varargin; |
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104 ## varargin{1} is an ODE options structure opt |
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105 odeopts = varargin{1}; |
22626 | 106 funarguments = {varargin{2:numel (varargin)}}; |
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107 else # if (isstruct (varargin{1})) |
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108 odeopts = varargin{1}; |
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109 funarguments = {}; |
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110 endif |
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111 else # nargin == 3 |
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112 odeopts = odeset (); |
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113 funarguments = {}; |
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114 endif |
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115 |
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116 if (! isnumeric (trange) || ! isvector (trange)) |
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117 error ("Octave:invalid-input-arg", |
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118 "ode45: TRANGE must be a numeric vector"); |
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119 endif |
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120 |
22626 | 121 if (numel (trange) < 2) |
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122 error ("Octave:invalid-input-arg", |
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123 "ode45: TRANGE must contain at least 2 elements"); |
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124 elseif (trange(1) == trange(2)) |
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125 error ("Octave:invalid-input-arg", |
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126 "ode45: invalid time span, TRANGE(1) == TRANGE(2)"); |
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127 else |
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128 direction = sign (trange(2) - trange(1)); |
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129 endif |
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130 trange = trange(:); |
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131 |
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132 if (! isnumeric (init) || ! isvector (init)) |
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133 error ("Octave:invalid-input-arg", |
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134 "ode45: INIT must be a numeric vector"); |
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135 endif |
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136 init = init(:); |
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137 |
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138 if (ischar (fun)) |
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139 try |
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140 fun = str2func (fun); |
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141 catch |
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142 warning (lasterr); |
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143 end_try_catch |
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144 endif |
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145 if (! isa (fun, "function_handle")) |
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146 error ("Octave:invalid-input-arg", |
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147 "ode45: FUN must be a valid function handle"); |
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148 endif |
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149 |
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150 ## Start preprocessing, have a look which options are set in odeopts, |
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151 ## check if an invalid or unused option is set |
22626 | 152 [defaults, classes, attributes] = odedefaults (numel (init), |
153 trange(1), trange(end)); | |
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154 |
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155 ## FIXME: Refine is not correctly implemented yet |
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156 defaults = odeset (defaults, "Refine", 4); |
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157 |
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158 persistent ode45_ignore_options = ... |
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159 {"BDF", "InitialSlope", "Jacobian", "JPattern", |
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160 "MassSingular", "MaxOrder", "MvPattern", "Vectorized"}; |
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161 |
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162 defaults = rmfield (defaults, ode45_ignore_options); |
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163 classes = rmfield (classes, ode45_ignore_options); |
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164 attributes = rmfield (attributes, ode45_ignore_options); |
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165 |
22626 | 166 odeopts = odemergeopts ("ode45", odeopts, defaults, classes, attributes); |
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167 |
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168 odeopts.funarguments = funarguments; |
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169 odeopts.direction = direction; |
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170 |
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171 if (! isempty (odeopts.NonNegative)) |
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172 if (isempty (odeopts.Mass)) |
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173 odeopts.havenonnegative = true; |
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174 else |
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175 odeopts.havenonnegative = false; |
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176 warning ("Octave:invalid-input-arg", |
22626 | 177 ['ode45: option "NonNegative" is ignored', ... |
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178 " when mass matrix is set\n"]); |
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179 endif |
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180 else |
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181 odeopts.havenonnegative = false; |
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182 endif |
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183 |
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184 if (isempty (odeopts.OutputFcn) && nargout == 0) |
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185 odeopts.OutputFcn = @odeplot; |
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186 odeopts.haveoutputfunction = true; |
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187 else |
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188 odeopts.haveoutputfunction = ! isempty (odeopts.OutputFcn); |
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189 endif |
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190 |
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191 if (isempty (odeopts.InitialStep)) |
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192 odeopts.InitialStep = odeopts.direction * ... |
22626 | 193 starting_stepsize (order, fun, trange(1), init, |
194 odeopts.AbsTol, odeopts.RelTol, | |
195 strcmp (odeopts.NormControl, "on"), | |
196 odeopts.funarguments); | |
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197 endif |
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198 |
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199 if (! isempty (odeopts.Mass) && isnumeric (odeopts.Mass)) |
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200 havemasshandle = false; |
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201 mass = odeopts.Mass; # constant mass |
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202 elseif (isa (odeopts.Mass, "function_handle")) |
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203 havemasshandle = true; # mass defined by a function handle |
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204 else # no mass matrix - creating a diag-matrix of ones for mass |
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205 havemasshandle = false; # mass = diag (ones (length (init), 1), 0); |
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206 endif |
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207 |
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208 ## Starting the initialization of the core solver ode45 |
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209 |
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210 if (havemasshandle) # Handle only the dynamic mass matrix, |
22626 | 211 if (! strcmp (odeopts.MStateDependence, "none")) |
212 ### constant mass matrices have already | |
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213 mass = @(t,x) odeopts.Mass (t, x, odeopts.funarguments{:}); |
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214 fun = @(t,x) mass (t, x, odeopts.funarguments{:}) ... |
22626 | 215 \ fun (t, x, odeopts.funarguments{:}); |
216 else | |
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217 mass = @(t) odeopts.Mass (t, odeopts.funarguments{:}); |
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218 fun = @(t,x) mass (t, odeopts.funarguments{:}) ... |
22626 | 219 \ fun (t, x, odeopts.funarguments{:}); |
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220 endif |
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221 endif |
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222 |
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223 if (nargout == 1) |
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224 ## Single output requires auto-selected intermediate times, |
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225 ## which is obtained by NOT specifying specific solution times. |
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226 trange = [trange(1); trange(end)]; |
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227 odeopts.Refine = []; # disable Refine when single output requested |
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228 elseif (numel (trange) > 2) |
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229 odeopts.Refine = []; # disable Refine when specific times requested |
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230 endif |
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231 |
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232 solution = integrate_adaptive (@runge_kutta_45_dorpri, |
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233 order, fun, trange, init, odeopts); |
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234 |
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235 ## Postprocessing, do whatever when terminating integration algorithm |
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236 if (odeopts.haveoutputfunction) # Cleanup plotter |
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237 feval (odeopts.OutputFcn, [], [], "done", odeopts.funarguments{:}); |
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238 endif |
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239 if (! isempty (odeopts.Events)) # Cleanup event function handling |
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241 solution.x(end,:).', "done", odeopts.funarguments{:}); |
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242 endif |
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243 |
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244 ## Print additional information if option Stats is set |
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245 if (strcmp (odeopts.Stats, "on")) |
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246 nsteps = solution.cntloop; # cntloop from 2..end |
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247 nfailed = solution.cntcycles - nsteps; # cntcycl from 1..end |
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248 nfevals = 6 * solution.cntcycles + 1; # number of ode evaluations |
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249 ndecomps = 0; # number of LU decompositions |
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250 npds = 0; # number of partial derivatives |
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251 nlinsols = 0; # no. of linear systems solutions |
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252 ## Print cost statistics if no output argument is given |
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253 if (nargout == 0) |
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254 printf ("Number of successful steps: %d\n", nsteps); |
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255 printf ("Number of failed attempts: %d\n", nfailed); |
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256 printf ("Number of function calls: %d\n", nfevals); |
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257 endif |
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258 endif |
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259 |
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260 if (nargout == 2) |
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261 varargout{1} = solution.t; # Time stamps are first output argument |
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262 varargout{2} = solution.x; # Results are second output argument |
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263 elseif (nargout == 1) |
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264 varargout{1}.x = solution.t.'; # Time stamps are saved in field x (row vector) |
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265 varargout{1}.y = solution.x.'; # Results are saved in field y (row vector) |
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266 varargout{1}.solver = solver; # Solver name is saved in field solver |
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267 if (! isempty (odeopts.Events)) |
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268 varargout{1}.ie = solution.event{2}; # Index info which event occurred |
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269 varargout{1}.xe = solution.event{3}; # Time info when an event occurred |
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270 varargout{1}.ye = solution.event{4}; # Results when an event occurred |
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271 endif |
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272 if (strcmp (odeopts.Stats, "on")) |
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273 varargout{1}.stats = struct (); |
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274 varargout{1}.stats.nsteps = nsteps; |
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275 varargout{1}.stats.nfailed = nfailed; |
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276 varargout{1}.stats.nfevals = nfevals; |
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277 varargout{1}.stats.npds = npds; |
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278 varargout{1}.stats.ndecomps = ndecomps; |
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279 varargout{1}.stats.nlinsols = nlinsols; |
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280 endif |
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281 elseif (nargout > 2) |
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282 varargout = cell (1,5); |
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283 varargout{1} = solution.t; |
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284 varargout{2} = solution.x; |
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285 if (! isempty (odeopts.Events)) |
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286 varargout{3} = solution.event{3}; # Time info when an event occurred |
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287 varargout{4} = solution.event{4}; # Results when an event occurred |
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288 varargout{5} = solution.event{2}; # Index info which event occurred |
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289 endif |
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290 endif |
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291 |
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292 endfunction |
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293 |
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294 |
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295 %!demo |
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296 %! ## Demonstrate convergence order for ode45 |
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297 %! tol = 1e-5 ./ 10.^[0:8]; |
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298 %! for i = 1 : numel (tol) |
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299 %! opt = odeset ("RelTol", tol(i), "AbsTol", realmin); |
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300 %! [t, y] = ode45 (@(t, y) -y, [0, 1], 1, opt); |
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301 %! h(i) = 1 / (numel (t) - 1); |
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302 %! err(i) = norm (y .* exp (t) - 1, Inf); |
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303 %! endfor |
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304 %! |
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305 %! ## Estimate order visually |
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306 %! loglog (h, tol, "-ob", |
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307 %! h, err, "-b", |
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308 %! h, (h/h(end)) .^ 4 .* tol(end), "k--", |
22626 | 309 %! h, (h/h(end)) .^ 5 .* tol(end), "k-"); |
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310 %! axis tight |
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311 %! xlabel ("h"); |
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312 %! ylabel ("err(h)"); |
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313 %! title ("Convergence plot for ode45"); |
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314 %! legend ("imposed tolerance", "ode45 (relative) error", |
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315 %! "order 4", "order 5", "location", "northwest"); |
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316 %! |
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317 %! ## Estimate order numerically |
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318 %! p = diff (log (err)) ./ diff (log (h)) |
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319 |
22626 | 320 ## We are using the Van der Pol equation for all tests that are done |
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321 ## for this function. |
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322 ## For further tests we also define a reference solution (computed at high |
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323 ## accuracy) |
22626 | 324 %!function ydot = fpol (t, y) # The Van der Pol ODE |
325 %! ydot = [y(2); (1 - y(1)^2) * y(2) - y(1)]; | |
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326 %!endfunction |
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327 %!function ref = fref () # The computed reference solution |
22626 | 328 %! ref = [0.32331666704577, -1.83297456798624]; |
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329 %!endfunction |
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330 %!function [val, trm, dir] = feve (t, y, varargin) |
22626 | 331 %! val = fpol (t, y, varargin); # We use the derivatives |
332 %! trm = zeros (2,1); # that's why component 2 | |
333 %! dir = ones (2,1); # does not seem to be exact | |
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334 %!endfunction |
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335 %!function [val, trm, dir] = fevn (t, y, varargin) |
22626 | 336 %! val = fpol (t, y, varargin); # We use the derivatives |
337 %! trm = ones (2,1); # that's why component 2 | |
338 %! dir = ones (2,1); # does not seem to be exact | |
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339 %!endfunction |
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340 %!function mas = fmas (t, y, varargin) |
22626 | 341 %! mas = [1, 0; 0, 1]; # Dummy mass matrix for tests |
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342 %!endfunction |
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343 %!function mas = fmsa (t, y, varargin) |
22626 | 344 %! mas = sparse ([1, 0; 0, 1]); # A sparse dummy matrix |
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345 %!endfunction |
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346 %!function out = fout (t, y, flag, varargin) |
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347 %! out = false; |
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348 %! if (strcmp (flag, "init")) |
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349 %! if (! isequal (size (t), [2, 1])) |
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350 %! error ('fout: step "init"'); |
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351 %! endif |
22626 | 352 %! elseif (isempty (flag)) |
22639
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353 %! if (! isequal (size (t), [1, 1])) |
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354 %! error ('fout: step "calc"'); |
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355 %! endif |
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356 %! elseif (strcmp (flag, "done")) |
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357 %! if (! isempty (t)) |
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358 %! warning ('fout: step "done"'); |
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359 %! endif |
22626 | 360 %! else |
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361 %! error ("fout: invalid flag <%s>", flag); |
22626 | 362 %! endif |
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363 %!endfunction |
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364 %! |
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365 %!test # two output arguments |
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366 %! [t, y] = ode45 (@fpol, [0 2], [2 0]); |
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367 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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368 %!test # not too many steps |
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369 %! [t, y] = ode45 (@fpol, [0 2], [2 0]); |
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370 %! assert (size (t) < 20); |
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371 %!test # anonymous function instead of real function |
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Cleanup variable names, obsolete ChangeLog info in ode23, ode45.
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372 %! fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)]; |
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373 %! [t, y] = ode45 (fvdp, [0 2], [2 0]); |
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374 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
20613
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375 %!test # string instead of function |
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376 %! [t, y] = ode45 ("fpol", [0 2], [2 0]); |
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377 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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378 %!test # extra input arguments passed through |
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379 %! [t, y] = ode45 (@fpol, [0 2], [2 0], 12, 13, "KL"); |
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380 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
20631
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381 %!test # empty ODEOPT structure *but* extra input arguments |
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382 %! opt = odeset; |
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383 %! [t, y] = ode45 (@fpol, [0 2], [2 0], opt, 12, 13, "KL"); |
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384 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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385 %!test # Solve another anonymous function below zero |
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386 %! vref = [0, 14.77810590694212]; |
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387 %! [t, y] = ode45 (@(t,y) y, [-2 0], 2); |
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388 %! assert ([t(end), y(end,:)], vref, 1e-1); |
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389 %!test # InitialStep option |
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390 %! opt = odeset ("InitialStep", 1e-8); |
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391 %! [t, y] = ode45 (@fpol, [0 0.2], [2 0], opt); |
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392 %! assert ([t(2)-t(1)], [1e-8], 1e-9); |
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393 %!test # MaxStep option |
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394 %! opt = odeset ("MaxStep", 1e-3); |
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395 %! sol = ode45 (@fpol, [0 0.2], [2 0], opt); |
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396 %! assert ([sol.x(5)-sol.x(4)], [1e-3], 1e-3); |
22659
04fc7e9c5f96
Disable Refine option of ode solvers for certain input/output combinations (partial fix bug #49408).
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397 %!test # Solve with intermediate step |
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398 %! [t, y] = ode45 (@fpol, [0 1 2], [2 0]); |
04fc7e9c5f96
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399 %! assert (any((t-1) == 0)); |
04fc7e9c5f96
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400 %! assert ([t(end), y(end,:)], [2, fref], 1e-3); |
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401 %!test # Solve in backward direction starting at t=0 |
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402 %! vref = [-1.205364552835178, 0.951542399860817]; |
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403 %! sol = ode45 (@fpol, [0 -2], [2 0]); |
22654
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Change orientation of output fields in struct returned from ode solvers (bug #49402).
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404 %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-2); |
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405 %!test # Solve in backward direction starting at t=2 |
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406 %! vref = [-1.205364552835178, 0.951542399860817]; |
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407 %! sol = ode45 (@fpol, [2 -2], fref); |
22654
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Change orientation of output fields in struct returned from ode solvers (bug #49402).
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408 %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-2); |
22659
04fc7e9c5f96
Disable Refine option of ode solvers for certain input/output combinations (partial fix bug #49408).
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409 %!test # Solve in backward direction starting at t=2, with intermediate step |
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410 %! vref = [-1.205364552835178, 0.951542399860817]; |
22659
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411 %! [t, y] = ode45 (@fpol, [2 0 -2], fref); |
04fc7e9c5f96
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412 %! idx = find(y < 0, 1, "first") - 1; |
04fc7e9c5f96
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413 %! assert ([t(idx), y(idx,:)], [0,2,0], 1e-2); |
04fc7e9c5f96
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414 %! assert ([t(end), y(end,:)], [-2, vref], 1e-2); |
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415 %!test # Solve another anonymous function in backward direction |
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416 %! vref = [-1, 0.367879437558975]; |
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417 %! sol = ode45 (@(t,y) y, [0 -1], 1); |
22654
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418 %! assert ([sol.x(end); sol.y(:,end)], vref', 1e-3); |
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419 %!test # Solve another anonymous function below zero |
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420 %! vref = [0, 14.77810590694212]; |
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421 %! sol = ode45 (@(t,y) y, [-2 0], 2); |
22654
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422 %! assert ([sol.x(end); sol.y(:,end)], vref', 1e-3); |
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423 %!test # Solve in backward direction starting at t=0 with MaxStep option |
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424 %! vref = [-1.205364552835178, 0.951542399860817]; |
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425 %! opt = odeset ("MaxStep", 1e-3); |
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426 %! sol = ode45 (@fpol, [0 -2], [2 0], opt); |
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427 %! assert ([abs(sol.x(8)-sol.x(7))], [1e-3], 1e-3); |
22654
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428 %! assert ([sol.x(end); sol.y(:,end)], [-2; vref'], 1e-3); |
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429 %!test # AbsTol option |
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430 %! opt = odeset ("AbsTol", 1e-5); |
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431 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
22654
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432 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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433 %!test # AbsTol and RelTol option |
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434 %! opt = odeset ("AbsTol", 1e-8, "RelTol", 1e-8); |
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435 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
22654
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436 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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437 %!test # RelTol and NormControl option -- higher accuracy |
20634
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438 %! opt = odeset ("RelTol", 1e-8, "NormControl", "on"); |
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439 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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440 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-5); |
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441 %!test # Keeps initial values while integrating |
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442 %! opt = odeset ("NonNegative", 2); |
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443 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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444 %! assert ([sol.x(end); sol.y(:,end)], [2; 2; 0], 0.5); |
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445 %!test # Details of OutputSel and Refine can't be tested |
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446 %! opt = odeset ("OutputFcn", @fout, "OutputSel", 1, "Refine", 5); |
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447 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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448 %!test # Stats must add further elements in sol |
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449 %! opt = odeset ("Stats", "on"); |
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450 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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451 %! assert (isfield (sol, "stats")); |
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452 %! assert (isfield (sol.stats, "nsteps")); |
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453 %!test # Events option add further elements in sol |
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454 %! opt = odeset ("Events", @feve); |
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455 %! sol = ode45 (@fpol, [0 10], [2 0], opt); |
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456 %! assert (isfield (sol, "ie")); |
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457 %! assert (sol.ie(1), 2); |
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458 %! assert (isfield (sol, "xe")); |
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459 %! assert (isfield (sol, "ye")); |
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460 %!test # Events option, now stop integration |
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461 %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); |
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462 %! opt = odeset ("Events", @fevn, "NormControl", "on"); |
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463 %! sol = ode45 (@fpol, [0 10], [2 0], opt); |
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464 %! assert ([sol.ie, sol.xe, sol.ye], |
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465 %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); |
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466 %!test # Events option, five output arguments |
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467 %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); |
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468 %! opt = odeset ("Events", @fevn, "NormControl", "on"); |
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469 %! [t, y, vxe, ye, vie] = ode45 (@fpol, [0 10], [2 0], opt); |
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470 %! assert ([vie, vxe, ye], |
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471 %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); |
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472 %!test # Mass option as function |
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473 %! opt = odeset ("Mass", @fmas); |
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474 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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475 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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476 %!test # Mass option as matrix |
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477 %! opt = odeset ("Mass", eye (2,2)); |
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478 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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479 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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480 %!test # Mass option as sparse matrix |
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481 %! opt = odeset ("Mass", sparse (eye (2,2))); |
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482 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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483 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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484 %!test # Mass option as function and sparse matrix |
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485 %! opt = odeset ("Mass", @fmsa); |
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486 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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487 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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488 %!test # Mass option as function and MStateDependence |
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489 %! opt = odeset ("Mass", @fmas, "MStateDependence", "strong"); |
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490 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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491 %! assert ([sol.x(end); sol.y(:,end)], [2; fref'], 1e-3); |
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492 |
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493 ## Note: The following options have no effect on this solver |
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494 ## therefore it makes no sense to test them here: |
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495 ## |
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496 ## "BDF" |
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497 ## "InitialSlope" |
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498 ## "JPattern" |
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499 ## "Jacobian" |
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500 ## "MassSingular" |
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501 ## "MaxOrder" |
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502 ## "MvPattern" |
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503 ## "Vectorized" |
22626 | 504 |
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505 %!test # Check that imaginary part of solution does not get inverted |
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506 %! sol = ode45 (@(x,y) 1, [0 1], 1i); |
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507 %! assert (imag (sol.y), ones (size (sol.y))) |
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508 %! [x, y] = ode45 (@(x,y) 1, [0 1], 1i); |
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509 %! assert (imag (y), ones (size (y))) |
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510 |
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511 %!error ode45 () |
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512 %!error ode45 (1) |
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513 %!error ode45 (1,2) |
22626 | 514 %!error <TRANGE must be a numeric> ode45 (@fpol, {[0 25]}, [3 15 1]) |
515 %!error <TRANGE must be a .* vector> ode45 (@fpol, [0 25; 25 0], [3 15 1]) | |
516 %!error <TRANGE must contain at least 2 elements> ode45 (@fpol, [1], [3 15 1]) | |
517 %!error <invalid time span> ode45 (@fpol, [1 1], [3 15 1]) | |
518 %!error <INIT must be a numeric> ode45 (@fpol, [0 25], {[3 15 1]}) | |
519 %!error <INIT must be a .* vector> ode45 (@fpol, [0 25], [3 15 1; 3 15 1]) | |
520 %!error <FUN must be a valid function handle> ode45 (1, [0 25], [3 15 1]) |