Mercurial > octave
annotate scripts/ode/ode45.m @ 21604:d7a268e68e69
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author | John W. Eaton <jwe@octave.org> |
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date | Mon, 11 Apr 2016 10:53:18 -0400 |
parents | f29d68e24c5a |
children | dcf8922b724b |
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1 ## Copyright (C) 2014, Jacopo Corno <jacopo.corno@gmail.com> |
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2 ## Copyright (C) 2013, Roberto Porcu' <roberto.porcu@polimi.it> |
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3 ## Copyright (C) 2006-2012, Thomas Treichl <treichl@users.sourceforge.net> |
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4 ## |
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5 ## This file is part of Octave. |
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6 ## |
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7 ## Octave is free software; you can redistribute it and/or modify it |
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8 ## under the terms of the GNU General Public License as published by |
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9 ## the Free Software Foundation; either version 3 of the License, or (at |
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10 ## your option) any later version. |
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11 ## |
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12 ## Octave is distributed in the hope that it will be useful, but |
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13 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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14 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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15 ## General Public License for more details. |
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16 ## |
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17 ## You should have received a copy of the GNU General Public License |
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18 ## along with Octave; see the file COPYING. If not, see |
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19 ## <http://www.gnu.org/licenses/>. |
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20 |
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21 ## -*- texinfo -*- |
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22 ## @deftypefn {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}) |
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23 ## @deftypefnx {} {[@var{t}, @var{y}] =} ode45 (@var{fun}, @var{trange}, @var{init}, @var{ode_opt}) |
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24 ## @deftypefnx {} {[@var{t}, @var{y}] =} ode45 (@dots{}, @var{par1}, @var{par2}, @dots{}) |
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25 ## @deftypefnx {} {[@var{t}, @var{y}, @var{te}, @var{ye}, @var{ie}] =} ode45 (@dots{}) |
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26 ## @deftypefnx {} {@var{solution} =} ode45 (@dots{}) |
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27 ## |
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28 ## Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) |
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29 ## with the well known explicit @nospell{Dormand-Prince} method of order 4. |
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30 ## |
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31 ## @var{fun} is a function handle, inline function, or string containing the |
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32 ## name of the function that defines the ODE: @code{y' = f(t,y)}. The function |
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33 ## must accept two inputs where the first is time @var{t} and the second is a |
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34 ## column vector of unknowns @var{y}. |
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35 ## |
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36 ## @var{trange} specifies the time interval over which the ODE will be |
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37 ## evaluated. Typically, it is a two-element vector specifying the initial and |
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38 ## final times (@code{[tinit, tfinal]}). If there are more than two elements |
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39 ## then the solution will also be evaluated at these intermediate time |
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40 ## instances unless the integrate function specified is |
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41 ## @command{integrate_n_steps}. |
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42 ## |
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43 ## By default, @code{ode45} uses an adaptive timestep with the |
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44 ## @code{integrate_adaptive} algorithm. The tolerance for the timestep |
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45 ## computation may be changed by using the option @qcode{"Tau"}, that has a |
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46 ## default value of @math{1e-6}. If the ODE option @qcode{"TimeStepSize"} is |
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47 ## not empty, then the stepper called will be @code{integrate_const}. If, in |
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48 ## addition, the option @qcode{"TimeStepNumber"} is also specified then the |
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49 ## integrate function @code{integrate_n_steps} will be used. |
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50 ## |
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51 ## @var{init} contains the initial value for the unknowns. If it is a row |
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52 ## vector then the solution @var{y} will be a matrix in which each column is |
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53 ## the solution for the corresponding initial value in @var{init}. |
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54 ## |
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55 ## The optional fourth argument @var{ode_opt} specifies non-default options to |
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56 ## the ODE solver. It is a structure generated by @code{odeset}. |
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57 ## |
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58 ## The function typically returns two outputs. Variable @var{t} is a |
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59 ## column vector and contains the times where the solution was found. The |
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60 ## output @var{y} is a matrix in which each column refers to a different |
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61 ## unknown of the problem and each row corresponds to a time in @var{t}. |
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62 ## |
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63 ## The output can also be returned as a structure @var{solution} which |
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64 ## has field @var{x} containing the time where the solution was evaluated and |
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65 ## field @var{y} containing the solution matrix for the times in @var{x}. |
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66 ## Use @code{fieldnames (@var{solution})} to see the other fields and |
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67 ## additional information returned. |
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68 ## |
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69 ## If using the @qcode{"Events"} option then three additional outputs may |
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70 ## be returned. @var{te} holds the time when an Event function returned a |
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71 ## zero. @var{ye} holds the value of the solution at time @var{te}. @var{ie} |
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72 ## contains an index indicating which Event function was triggered in the case |
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73 ## of multiple Event functions. |
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74 ## |
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75 ## Example: Solve the @nospell{Van der Pol} equation |
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76 ## |
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77 ## @example |
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78 ## @group |
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79 ## fvdp = @@(@var{t},@var{y}) [@var{y}(2); (1 - @var{y}(1)^2) * @var{y}(2) - @var{y}(1)]; |
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80 ## [@var{t},@var{y}] = ode45 (fvdp, [0, 20], [2, 0]); |
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81 ## @end group |
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82 ## @end example |
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83 ## @seealso{odeset, odeget} |
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84 ## @end deftypefn |
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85 |
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86 function varargout = ode45 (fun, trange, init, varargin) |
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87 |
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88 if (nargin < 3) |
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89 print_usage (); |
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90 endif |
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91 |
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92 order = 5; # runge_kutta_45_dorpri uses local extrapolation |
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93 solver = "ode45"; |
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94 |
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95 if (nargin >= 4) |
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96 if (! isstruct (varargin{1})) |
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97 ## varargin{1:len} are parameters for fun |
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98 odeopts = odeset (); |
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99 odeopts.funarguments = varargin; |
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100 elseif (length (varargin) > 1) |
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101 ## varargin{1} is an ODE options structure vopt |
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102 odeopts = ode_struct_value_check ("ode45", varargin{1}, "ode45"); |
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103 odeopts.funarguments = {varargin{2:length(varargin)}}; |
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104 else # if (isstruct (varargin{1})) |
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105 odeopts = ode_struct_value_check ("ode45", varargin{1}, "ode45"); |
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106 odeopts.funarguments = {}; |
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107 endif |
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108 else # nargin == 3 |
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109 odeopts = odeset (); |
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110 odeopts.funarguments = {}; |
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111 endif |
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112 |
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113 if (! isnumeric (trange) || ! isvector (trange)) |
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114 error ("Octave:invalid-input-arg", |
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115 "ode45: TRANGE must be a numeric vector"); |
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116 endif |
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117 |
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118 TimeStepNumber = odeget (odeopts, "TimeStepNumber", [], "fast"); |
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119 TimeStepSize = odeget (odeopts, "TimeStepSize", [], "fast"); |
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120 if (length (trange) < 2 |
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121 && (isempty (TimeStepSize) || isempty (TimeStepNumber))) |
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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 odeopts.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 |
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152 if (isempty (TimeStepNumber) && isempty (TimeStepSize)) |
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153 integrate_func = "adaptive"; |
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154 odeopts.stepsizefixed = false; |
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155 elseif (! isempty (TimeStepNumber) && ! isempty (TimeStepSize)) |
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156 integrate_func = "n_steps"; |
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157 odeopts.stepsizefixed = true; |
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158 if (sign (TimeStepSize) != odeopts.direction) |
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159 warning ("Octave:invalid-input-arg", |
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160 ["ode45: option 'TimeStepSize' has the wrong sign, ", ... |
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161 "but will be corrected automatically\n"]); |
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162 TimeStepSize = -TimeStepSize; |
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163 endif |
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164 elseif (isempty (TimeStepNumber) && ! isempty (TimeStepSize)) |
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165 integrate_func = "const"; |
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166 odeopts.stepsizefixed = true; |
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167 if (sign (TimeStepSize) != odeopts.direction) |
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168 warning ("Octave:invalid-input-arg", |
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169 ["ode45: option 'TimeStepSize' has the wrong sign, ", |
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170 "but will be corrected automatically\n"]); |
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171 TimeStepSize = -TimeStepSize; |
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172 endif |
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173 else |
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174 warning ("Octave:invalid-input-arg", |
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175 "ode45: assuming an adaptive integrate function\n"); |
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176 integrate_func = "adaptive"; |
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177 endif |
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178 |
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179 if (isempty (odeopts.RelTol) && ! odeopts.stepsizefixed) |
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180 odeopts.RelTol = 1e-3; |
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181 elseif (! isempty (odeopts.RelTol) && odeopts.stepsizefixed) |
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182 warning ("Octave:invalid-input-arg", |
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183 ["ode45: option 'RelTol' is ignored", ... |
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184 " when fixed time stamps are given\n"]); |
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185 endif |
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186 |
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187 if (isempty (odeopts.AbsTol) && ! odeopts.stepsizefixed) |
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188 odeopts.AbsTol = 1e-6; |
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189 elseif (! isempty (odeopts.AbsTol) && odeopts.stepsizefixed) |
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190 warning ("Octave:invalid-input-arg", |
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191 ["ode45: option 'AbsTol' is ignored", ... |
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192 " when fixed time stamps are given\n"]); |
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193 else |
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194 odeopts.AbsTol = odeopts.AbsTol(:); # Create column vector |
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195 endif |
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196 |
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197 odeopts.normcontrol = strcmp (odeopts.NormControl, "on"); |
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198 |
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199 if (! isempty (odeopts.NonNegative)) |
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200 if (isempty (odeopts.Mass)) |
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201 odeopts.havenonnegative = true; |
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202 else |
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203 odeopts.havenonnegative = false; |
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204 warning ("Octave:invalid-input-arg", |
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205 ["ode45: option 'NonNegative' is ignored", ... |
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206 " when mass matrix is set\n"]); |
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207 endif |
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208 else |
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209 odeopts.havenonnegative = false; |
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210 endif |
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211 |
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212 if (isempty (odeopts.OutputFcn) && nargout == 0) |
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213 odeopts.OutputFcn = @odeplot; |
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214 odeopts.haveoutputfunction = true; |
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215 else |
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216 odeopts.haveoutputfunction = ! isempty (odeopts.OutputFcn); |
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217 endif |
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218 |
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219 odeopts.haveoutputselection = ! isempty (odeopts.OutputSel); |
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220 |
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221 if (odeopts.Refine > 0) |
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222 odeopts.haverefine = true; |
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223 else |
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224 odeopts.haverefine = false; |
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225 endif |
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226 |
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227 if (isempty (odeopts.InitialStep) && strcmp (integrate_func, "adaptive")) |
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228 odeopts.InitialStep = ... |
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229 odeopts.direction * starting_stepsize (order, fun, trange(1), |
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230 init, |
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231 odeopts.AbsTol, |
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232 odeopts.RelTol, |
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233 odeopts.normcontrol); |
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234 elseif (isempty (odeopts.InitialStep)) |
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235 odeopts.InitialStep = TimeStepSize; |
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236 endif |
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237 |
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238 if (isempty (odeopts.MaxStep) && ! odeopts.stepsizefixed) |
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239 odeopts.MaxStep = abs (trange(end) - trange(1)) / 10; |
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240 endif |
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241 |
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242 odeopts.haveeventfunction = ! isempty (odeopts.Events); |
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243 |
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244 ## The options "Jacobian", "JPattern" and "Vectorized" will be ignored |
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245 ## by this solver because this solver uses an explicit Runge-Kutta method |
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246 ## and therefore no Jacobian calculation is necessary. |
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247 if (! isempty (odeopts.Jacobian)) |
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248 warning ("Octave:invalid-input-arg", |
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249 "ode45: option 'Jacobian' is ignored by this solver\n"); |
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250 endif |
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251 |
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252 if (! isempty (odeopts.JPattern)) |
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253 warning ("Octave:invalid-input-arg", |
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254 "ode45: option 'JPattern' is ignored by this solver\n"); |
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255 endif |
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256 |
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257 if (! isempty (odeopts.Vectorized)) |
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258 warning ("Octave:invalid-input-arg", |
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259 "ode45: option 'Vectorized' is ignored by this solver\n"); |
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260 endif |
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261 |
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262 if (! isempty (odeopts.Mass) && isnumeric (odeopts.Mass)) |
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263 havemasshandle = false; |
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264 mass = odeopts.Mass; # constant mass |
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265 elseif (isa (odeopts.Mass, "function_handle")) |
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266 havemasshandle = true; # mass defined by a function handle |
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267 else # no mass matrix - creating a diag-matrix of ones for mass |
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268 havemasshandle = false; # mass = diag (ones (length (init), 1), 0); |
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269 endif |
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270 |
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271 massdependence = ! strcmp (odeopts.MStateDependence, "none"); |
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272 |
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273 ## Other options that are not used by this solver. |
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274 if (! isempty (odeopts.MvPattern)) |
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275 warning ("Octave:invalid-input-arg", |
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276 "ode45: option 'MvPattern' is ignored by this solver\n"); |
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277 endif |
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278 |
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279 if (! isempty (odeopts.MassSingular)) |
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280 warning ("Octave:invalid-input-arg", |
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281 "ode45: option 'MassSingular' is ignored by this solver\n"); |
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282 endif |
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283 |
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284 if (! isempty (odeopts.InitialSlope)) |
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285 warning ("Octave:invalid-input-arg", |
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286 "ode45: option 'InitialSlope' is ignored by this solver\n"); |
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287 endif |
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288 |
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289 if (! isempty (odeopts.MaxOrder)) |
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290 warning ("Octave:invalid-input-arg", |
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291 "ode45: option 'MaxOrder' is ignored by this solver\n"); |
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292 endif |
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293 |
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294 if (! isempty (odeopts.BDF)) |
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295 warning ("Octave:invalid-input-arg", |
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296 "ode45: option 'BDF' is ignored by this solver\n"); |
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297 endif |
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298 |
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299 ## Starting the initialization of the core solver ode45 |
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300 |
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301 if (havemasshandle) # Handle only the dynamic mass matrix, |
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302 if (massdependence) # constant mass matrices have already |
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303 mass = @(t,x) odeopts.Mass (t, x, odeopts.funarguments{:}); |
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304 fun = @(t,x) mass (t, x, odeopts.funarguments{:}) ... |
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305 \ fun (t, x, odeopts.funarguments{:}); |
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306 else # if (massdependence == false) |
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307 mass = @(t) odeopts.Mass (t, odeopts.funarguments{:}); |
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308 fun = @(t,x) mass (t, odeopts.funarguments{:}) ... |
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309 \ fun (t, x, odeopts.funarguments{:}); |
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310 endif |
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311 endif |
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312 |
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313 switch (integrate_func) |
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314 case "adaptive" |
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315 solution = integrate_adaptive (@runge_kutta_45_dorpri, |
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316 order, fun, trange, init, odeopts); |
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317 case "n_steps" |
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318 solution = integrate_n_steps (@runge_kutta_45_dorpri, |
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319 fun, trange(1), init, |
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320 TimeStepSize, TimeStepNumber, odeopts); |
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321 case "const" |
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322 solution = integrate_const (@runge_kutta_45_dorpri, |
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323 fun, trange, init, |
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324 TimeStepSize, odeopts); |
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325 endswitch |
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326 |
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327 ## Postprocessing, do whatever when terminating integration algorithm |
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328 if (odeopts.haveoutputfunction) # Cleanup plotter |
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329 feval (odeopts.OutputFcn, solution.t(end), |
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330 solution.x(end,:)', "done", odeopts.funarguments{:}); |
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331 endif |
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332 if (odeopts.haveeventfunction) # Cleanup event function handling |
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333 ode_event_handler (odeopts.Events, solution.t(end), |
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334 solution.x(end,:)', "done", |
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335 odeopts.funarguments{:}); |
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336 endif |
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337 |
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338 ## Print additional information if option Stats is set |
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339 if (strcmp (odeopts.Stats, "on")) |
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340 havestats = true; |
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341 nsteps = solution.cntloop-2; # cntloop from 2..end |
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342 nfailed = (solution.cntcycles-1)-(nsteps)+1; # cntcycl from 1..end |
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343 nfevals = 6 * (solution.cntcycles-1) + 1; # number of ode evaluations |
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344 ndecomps = 0; # number of LU decompositions |
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345 npds = 0; # number of partial derivatives |
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346 nlinsols = 0; # no. of linear systems solutions |
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347 ## Print cost statistics if no output argument is given |
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348 if (nargout == 0) |
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349 printf ("Number of successful steps: %d\n", nsteps); |
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350 printf ("Number of failed attempts: %d\n", nfailed); |
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351 printf ("Number of function calls: %d\n", nfevals); |
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352 endif |
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353 else |
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354 havestats = false; |
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355 endif |
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356 |
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357 if (nargout == 2) |
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358 varargout{1} = solution.t; # Time stamps are first output argument |
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359 varargout{2} = solution.x; # Results are second output argument |
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360 elseif (nargout == 1) |
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361 varargout{1}.x = solution.t; # Time stamps are saved in field x |
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362 varargout{1}.y = solution.x; # Results are saved in field y |
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363 varargout{1}.solver = solver; # Solver name is saved in field solver |
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364 if (odeopts.haveeventfunction) |
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365 varargout{1}.ie = solution.event{2}; # Index info which event occurred |
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366 varargout{1}.xe = solution.event{3}; # Time info when an event occurred |
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367 varargout{1}.ye = solution.event{4}; # Results when an event occurred |
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368 endif |
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369 if (havestats) |
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370 varargout{1}.stats = struct (); |
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371 varargout{1}.stats.nsteps = nsteps; |
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372 varargout{1}.stats.nfailed = nfailed; |
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373 varargout{1}.stats.nfevals = nfevals; |
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374 varargout{1}.stats.npds = npds; |
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375 varargout{1}.stats.ndecomps = ndecomps; |
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376 varargout{1}.stats.nlinsols = nlinsols; |
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377 endif |
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378 elseif (nargout == 5) |
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379 varargout = cell (1,5); |
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380 varargout{1} = solution.t; |
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381 varargout{2} = solution.x; |
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382 if (odeopts.haveeventfunction) |
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383 varargout{3} = solution.event{3}; # Time info when an event occurred |
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384 varargout{4} = solution.event{4}; # Results when an event occurred |
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385 varargout{5} = solution.event{2}; # Index info which event occurred |
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386 endif |
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387 endif |
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388 |
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389 endfunction |
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390 |
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391 |
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392 %!demo |
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393 %! |
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394 %! ## Demonstrate convergence order for ode45 |
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395 %! tol = 1e-5 ./ 10.^[0:8]; |
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396 %! for i = 1 : numel (tol) |
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397 %! opt = odeset ("RelTol", tol(i), "AbsTol", realmin); |
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398 %! [t, y] = ode45 (@(t, y) -y, [0, 1], 1, opt); |
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399 %! h(i) = 1 / (numel (t) - 1); |
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400 %! err(i) = norm (y .* exp (t) - 1, Inf); |
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401 %! endfor |
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402 %! |
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403 %! ## Estimate order numerically |
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404 %! p = diff (log (err)) ./ diff (log (h)) |
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405 %! |
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406 %! ## Estimate order visually |
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407 %! figure (); |
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408 %! loglog (h, tol, "-ob", |
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409 %! h, err, "-b" |
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410 %! h, (h/h(end)) .^ 4 .* tol(end), "k--", |
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411 %! h, (h/h(end)) .^ 5 .* tol(end), "k-") |
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412 %! axis tight |
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413 %! xlabel ("h"); |
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414 %! ylabel ("err(h)"); |
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415 %! title ("Convergence plot for ode45"); |
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416 %! legend ("imposed tolerance", "ode45 (relative) error", |
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417 %! "order 4", "order 5", "location", "northwest"); |
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418 |
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419 ## We are using the "Van der Pol" implementation for all tests that are done |
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420 ## for this function. |
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421 ## For further tests we also define a reference solution (computed at high |
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422 ## accuracy) |
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423 %!function ydot = fpol (t, y) # The Van der Pol |
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424 %! ydot = [y(2); (1 - y(1)^2) * y(2) - y(1)]; |
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Moving ode45, odeset, odeget, and levenshtein from odepkg to core.
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425 %!endfunction |
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426 %!function ref = fref () # The computed reference solution |
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427 %! ref = [0.32331666704577, -1.83297456798624]; |
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428 %!endfunction |
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429 %!function jac = fjac (t, y, varargin) # its Jacobian |
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430 %! jac = [0, 1; -1 - 2 * y(1) * y(2), 1 - y(1)^2]; |
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431 %!endfunction |
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432 %!function jac = fjcc (t, y, varargin) # sparse type |
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433 %! jac = sparse ([0, 1; -1 - 2 * y(1) * y(2), 1 - y(1)^2]); |
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434 %!endfunction |
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435 %!function [val, trm, dir] = feve (t, y, varargin) |
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436 %! val = fpol (t, y, varargin); # We use the derivatives |
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437 %! trm = zeros (2,1); # that's why component 2 |
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438 %! dir = ones (2,1); # seems to not be exact |
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439 %!endfunction |
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440 %!function [val, trm, dir] = fevn (t, y, varargin) |
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441 %! val = fpol (t, y, varargin); # We use the derivatives |
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442 %! trm = ones (2,1); # that's why component 2 |
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443 %! dir = ones (2,1); # seems to not be exact |
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444 %!endfunction |
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445 %!function mas = fmas (t, y, varargin) |
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446 %! mas = [1, 0; 0, 1]; # Dummy mass matrix for tests |
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447 %!endfunction |
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448 %!function mas = fmsa (t, y, varargin) |
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449 %! mas = sparse ([1, 0; 0, 1]); # A sparse dummy matrix |
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450 %!endfunction |
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451 %!function out = fout (t, y, flag, varargin) |
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452 %! if (regexp (char (flag), 'init') == 1) |
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453 %! if (any (size (t) != [2, 1])) error ('"fout" step "init"'); endif |
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454 %! elseif (isempty (flag)) |
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455 %! if (any (size (t) != [1, 1])) error ('"fout" step "calc"'); endif |
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456 %! out = false; |
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457 %! elseif (regexp (char (flag), 'done') == 1) |
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458 %! if (any (size (t) != [1, 1])) error ('"fout" step "done"'); endif |
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459 %! else |
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460 %! error ('"fout" invalid flag'); |
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461 %! endif |
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462 %!endfunction |
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jcorno <jacopo.corno@gmail.com>
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463 %! |
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464 %!test # two output arguments |
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465 %! [t, y] = ode45 (@fpol, [0 2], [2 0]); |
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466 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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467 %!test # not too many steps |
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468 %! [t, y] = ode45 (@fpol, [0 2], [2 0]); |
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469 %! assert (size (t) < 20); |
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470 %!test # anonymous function instead of real function |
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471 %! fvdb = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)]; |
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472 %! [t, y] = ode45 (fvdb, [0 2], [2 0]); |
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473 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
20613
27c091f4b66d
allow first argument in ode45 to be a string
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474 %!test # string instead of function |
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475 %! [t, y] = ode45 ("fpol", [0 2], [2 0]); |
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476 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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477 %!test # extra input arguments passed through |
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478 %! [t, y] = ode45 (@fpol, [0 2], [2 0], 12, 13, "KL"); |
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479 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
20631
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maint: Remove obsolete ODE options from odeset/odeget/ode45.
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480 %!test # empty ODEOPT structure *but* extra input arguments |
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481 %! opt = odeset; |
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482 %! [t, y] = ode45 (@fpol, [0 2], [2 0], opt, 12, 13, "KL"); |
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483 %! assert ([t(end), y(end,:)], [2, fref], 1e-2); |
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484 %!test # Solve vdp in fixed step sizes |
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485 %! opt = odeset("TimeStepSize", 0.1); |
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486 %! [t, y] = ode45 (@fpol, [0,2], [2 0], opt); |
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487 %! assert (t(:), [0:0.1:2]', 1e-2); |
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488 %!test # Solve another anonymous function below zero |
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489 %! vref = [0, 14.77810590694212]; |
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490 %! [t, y] = ode45 (@(t,y) y, [-2 0], 2); |
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491 %! assert ([t(end), y(end,:)], vref, 1e-1); |
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492 %!test # InitialStep option |
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493 %! opt = odeset ("InitialStep", 1e-8); |
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494 %! [t, y] = ode45 (@fpol, [0 0.2], [2 0], opt); |
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495 %! assert ([t(2)-t(1)], [1e-8], 1e-9); |
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496 %!test # MaxStep option |
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497 %! opt = odeset ("MaxStep", 1e-3); |
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498 %! sol = ode45 (@fpol, [0 0.2], [2 0], opt); |
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499 %! assert ([sol.x(5)-sol.x(4)], [1e-3], 1e-3); |
20549
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500 %!test # Solve with intermidiate step |
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501 %! sol = ode45 (@fpol, [0 1 2], [2 0]); |
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502 %! assert (any((sol.x-1) == 0)); |
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503 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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504 %!test # Solve in backward direction starting at t=0 |
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505 %! vref = [-1.205364552835178, 0.951542399860817]; |
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506 %! sol = ode45 (@fpol, [0 -2], [2 0]); |
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507 %! assert ([sol.x(end), sol.y(end,:)], [-2, vref], 1e-2); |
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508 %!test # Solve in backward direction starting at t=2 |
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509 %! vref = [-1.205364552835178, 0.951542399860817]; |
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510 %! sol = ode45 (@fpol, [2 -2], fref); |
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511 %! assert ([sol.x(end), sol.y(end,:)], [-2, vref], 1e-2); |
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512 %!test # Solve in backward direction starting at t=2, with intermidiate step |
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513 %! vref = [-1.205364552835178, 0.951542399860817]; |
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514 %! sol = ode45 (@fpol, [2 0 -2], fref); |
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515 %! idx = find(sol.x < 0, 1, "first") - 1; |
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516 %! assert ([sol.x(idx), sol.y(idx,:)], [0 2 0], 1e-2); |
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517 %! assert ([sol.x(end), sol.y(end,:)], [-2, vref], 1e-2); |
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518 %!test # Solve another anonymous function in backward direction |
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519 %! vref = [-1, 0.367879437558975]; |
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520 %! sol = ode45 (@(t,y) y, [0 -1], 1); |
80e630b37ba1
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521 %! assert ([sol.x(end), sol.y(end,:)], vref, 1e-3); |
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522 %!test # Solve another anonymous function below zero |
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523 %! vref = [0, 14.77810590694212]; |
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524 %! sol = ode45 (@(t,y) y, [-2 0], 2); |
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525 %! assert ([sol.x(end), sol.y(end,:)], vref, 1e-3); |
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526 %!test # Solve in backward direction starting at t=0 with MaxStep option |
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527 %! vref = [-1.205364552835178, 0.951542399860817]; |
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528 %! opt = odeset ("MaxStep", 1e-3); |
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529 %! sol = ode45 (@fpol, [0 -2], [2 0], opt); |
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530 %! assert ([abs(sol.x(8)-sol.x(7))], [1e-3], 1e-3); |
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531 %! assert ([sol.x(end), sol.y(end,:)], [-2, vref], 1e-3); |
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532 %!test # AbsTol option |
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533 %! opt = odeset ("AbsTol", 1e-5); |
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534 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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535 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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536 %!test # AbsTol and RelTol option |
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537 %! opt = odeset ("AbsTol", 1e-8, "RelTol", 1e-8); |
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538 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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539 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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540 %!test # RelTol and NormControl option -- higher accuracy |
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541 %! opt = odeset ("RelTol", 1e-8, "NormControl", "on"); |
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542 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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543 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-5); |
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544 %!test # Keeps initial values while integrating |
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545 %! opt = odeset ("NonNegative", 2); |
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546 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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547 %! assert ([sol.x(end), sol.y(end,:)], [2, 2, 0], 0.5); |
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548 %!test # Details of OutputSel and Refine can't be tested |
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549 %! opt = odeset ("OutputFcn", @fout, "OutputSel", 1, "Refine", 5); |
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550 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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551 %!test # Stats must add further elements in sol |
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552 %! opt = odeset ("Stats", "on"); |
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553 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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554 %! assert (isfield (sol, "stats")); |
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555 %! assert (isfield (sol.stats, "nsteps")); |
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556 %!test # Events option add further elements in sol |
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557 %! opt = odeset ("Events", @feve); |
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558 %! sol = ode45 (@fpol, [0 10], [2 0], opt); |
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559 %! assert (isfield (sol, "ie")); |
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560 %! assert (sol.ie(1), 2); |
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561 %! assert (isfield (sol, "xe")); |
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562 %! assert (isfield (sol, "ye")); |
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563 %!test # Events option, now stop integration |
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564 %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); |
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565 %! opt = odeset ("Events", @fevn, "NormControl", "on"); |
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566 %! sol = ode45 (@fpol, [0 10], [2 0], opt); |
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567 %! assert ([sol.ie, sol.xe, sol.ye], |
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568 %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); |
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569 %!test # Events option, five output arguments |
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570 %! warning ("off", "integrate_adaptive:unexpected_termination", "local"); |
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571 %! opt = odeset ("Events", @fevn, "NormControl", "on"); |
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572 %! [t, y, vxe, ye, vie] = ode45 (@fpol, [0 10], [2 0], opt); |
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573 %! assert ([vie, vxe, ye], |
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574 %! [2.0, 2.496110, -0.830550, -2.677589], 6e-1); |
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575 %!test # Mass option as function |
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576 %! opt = odeset ("Mass", @fmas); |
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577 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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578 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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579 %!test # Mass option as matrix |
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580 %! opt = odeset ("Mass", eye (2,2)); |
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581 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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582 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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583 %!test # Mass option as sparse matrix |
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584 %! opt = odeset ("Mass", sparse (eye (2,2))); |
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585 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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586 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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587 %!test # Mass option as function and sparse matrix |
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588 %! opt = odeset ("Mass", @fmsa); |
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589 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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590 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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591 %!test # Mass option as function and MStateDependence |
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592 %! opt = odeset ("Mass", @fmas, "MStateDependence", "strong"); |
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593 %! sol = ode45 (@fpol, [0 2], [2 0], opt); |
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594 %! assert ([sol.x(end), sol.y(end,:)], [2, fref], 1e-3); |
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595 |
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596 %!error ode45 () |
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597 %!error ode45 (1) |
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598 %!error ode45 (1,2) |
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599 %!error <TRANGE must be a numeric> |
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600 %! ode45 (@fpol, {[0 25]}, [3 15 1]); |
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601 %!error <TRANGE must be a .* vector> |
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602 %! ode45 (@fpol, [0 25; 25 0], [3 15 1]); |
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603 %!error <TRANGE must contain at least 2 elements> |
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604 %! ode45 (@fpol, [1], [3 15 1]); |
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605 %!error <invalid time span> |
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606 %! ode45 (@fpol, [1 1], [3 15 1]); |
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607 %!error <INIT must be a numeric> |
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608 %! ode45 (@fpol, [0 25], {[3 15 1]}); |
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609 %!error <INIT must be a .* vector> |
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610 %! ode45 (@fpol, [0 25], [3 15 1; 3 15 1]); |
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611 %!error <FUN must be a valid function handle> |
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612 %! ode45 (1, [0 25], [3 15 1]); |
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613 %!error # strange ODEOPT structure |
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614 %! opt = struct ("foo", 1); |
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615 %! [t, y] = ode45 (@fpol, [0 2], [2 0], opt); |
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616 |