Mercurial > octave
annotate doc/interpreter/diffeq.txi @ 20901:afe9c529760d
2015 Code Sprint: move ode23 and runge_kutta_23 from odepkg to core
* scripts/ode/ode23.m: new file
* scripts/ode/private/runge_kutta_23.m: new file
* scripts/ode/module.mk: list new files
* doc/interpreter/diffeq.txi: mention ode23 among available solvers
* scripts/help/__unimplemented__.m: remove ode23 from list of unimplemented functions
| author | Stefan Miereis <stefan.miereis@gmx.de> |
|---|---|
| date | Tue, 15 Dec 2015 13:59:17 +0100 |
| parents | 75d09edcfc75 |
| children | dba88797f69f |
| rev | line source |
|---|---|
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1 @c Copyright (C) 1996-2015 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
|
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9 @c |
| 7018 | 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
|
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14 @c |
| 7018 | 15 @c You should have received a copy of the GNU General Public License |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Differential Equations |
| 3294 | 20 @chapter Differential Equations |
| 21 | |
| 6700 | 22 Octave has built-in functions for solving ordinary differential equations, |
| 23 and differential-algebraic equations. | |
| 8828 | 24 All solvers are based on reliable ODE routines written in Fortran. |
| 3294 | 25 |
| 26 @menu | |
|
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27 * Ordinary Differential Equations:: |
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28 * Differential-Algebraic Equations:: |
| 3294 | 29 @end menu |
| 30 | |
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31 @cindex differential equations |
| 3294 | 32 @cindex ODE |
| 33 @cindex DAE | |
| 34 | |
| 4167 | 35 @node Ordinary Differential Equations |
| 3294 | 36 @section Ordinary Differential Equations |
| 37 | |
| 38 The function @code{lsode} can be used to solve ODEs of the form | |
| 39 @tex | |
| 40 $$ | |
| 41 {dx\over dt} = f (x, t) | |
| 42 $$ | |
| 43 @end tex | |
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44 @ifnottex |
| 3294 | 45 |
| 46 @example | |
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47 @group |
| 3294 | 48 dx |
| 49 -- = f (x, t) | |
| 50 dt | |
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51 @end group |
| 3294 | 52 @end example |
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53 |
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54 @end ifnottex |
| 3294 | 55 |
| 56 @noindent | |
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57 using @nospell{Hindmarsh's} ODE solver @sc{lsode}. |
| 3294 | 58 |
| 6700 | 59 |
| 60 | |
| 3373 | 61 @DOCSTRING(lsode) |
| 3294 | 62 |
| 4115 | 63 @DOCSTRING(lsode_options) |
| 64 | |
| 3294 | 65 Here is an example of solving a set of three differential equations using |
| 66 @code{lsode}. Given the function | |
| 67 | |
| 68 @cindex oregonator | |
| 69 | |
| 70 @example | |
| 71 @group | |
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72 ## oregonator differential equation |
| 3294 | 73 function xdot = f (x, t) |
| 74 | |
| 75 xdot = zeros (3,1); | |
| 76 | |
| 77 xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ | |
| 78 - 8.375e-06*x(1)^2); | |
| 79 xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; | |
| 80 xdot(3) = 0.161*(x(1) - x(3)); | |
| 81 | |
| 82 endfunction | |
| 83 @end group | |
| 84 @end example | |
| 85 | |
| 86 @noindent | |
| 87 and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of | |
| 88 equations can be integrated using the command | |
| 89 | |
| 90 @example | |
| 91 @group | |
| 92 t = linspace (0, 500, 1000); | |
| 93 | |
| 94 y = lsode ("f", x0, t); | |
| 95 @end group | |
| 96 @end example | |
| 97 | |
| 98 If you try this, you will see that the value of the result changes | |
| 99 dramatically between @var{t} = 0 and 5, and again around @var{t} = 305. | |
| 100 A more efficient set of output points might be | |
| 101 | |
| 102 @example | |
| 103 @group | |
|
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104 t = [0, logspace(-1, log10(303), 150), \ |
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105 logspace(log10(304), log10(500), 150)]; |
| 3294 | 106 @end group |
| 107 @end example | |
| 108 | |
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109 See @nospell{Alan C. Hindmarsh}, |
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110 @cite{ODEPACK, A Systematized Collection of ODE Solvers}, |
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111 in Scientific Computing, @nospell{R. S. Stepleman}, editor, (1983) |
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112 for more information about the inner workings of @code{lsode}. |
| 3294 | 113 |
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114 An m-file for the differential equation used above is included with the |
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115 Octave distribution in the examples directory under the name |
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116 @file{oregonator.m}. |
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117 |
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118 @menu |
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119 * Matlab-compatible solvers:: |
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120 @end menu |
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121 |
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122 @node Matlab-compatible solvers |
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123 @subsection Matlab-compatible solvers |
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124 |
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125 Octave also provides a set of solvers for initial value problems for Ordinary |
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126 Differential Equations that have a @sc{matlab}-compatible interface. |
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127 The options for this class of methods are set using the functions. |
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128 |
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129 @itemize |
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130 @item @code{odeset} |
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131 |
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132 @item @code{odeget} |
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133 @end itemize |
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134 |
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135 Currently implemented solvers are: |
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136 |
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137 @itemize |
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138 @item @nospell{Runge-Kutta} methods |
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139 |
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140 @itemize |
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141 @item @code{ode45} Integrates a system of non--stiff ordinary differential equations |
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142 (non--stiff ODEs and DAEs) using second order @nospell{Dormand-Prince} |
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143 method. This is a fourth--order accurate integrator therefore the local |
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144 error normally expected is @math{O(h^5)}. This solver requires six |
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145 function evaluations per integration step. |
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146 @item @code{ode23} Integrates a system of non--stiff ordinary differential equations |
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147 (non-stiff ODEs and DAEs) using second order @nospell{Bogacki-Shampine} |
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148 method. This is a second-order accurate integrator therefore the local |
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149 error normally expected is @math{O(h^3)}. This solver requires three |
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150 function evaluations per integration step. |
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151 @end itemize |
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152 @end itemize |
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153 |
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154 |
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155 @DOCSTRING(ode45) |
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156 |
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157 @DOCSTRING(ode23) |
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158 |
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159 @DOCSTRING(odeset) |
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160 |
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161 @DOCSTRING(odeget) |
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162 |
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163 @DOCSTRING(odeplot) |
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164 |
| 4167 | 165 @node Differential-Algebraic Equations |
| 3294 | 166 @section Differential-Algebraic Equations |
| 167 | |
| 4115 | 168 The function @code{daspk} can be used to solve DAEs of the form |
| 3294 | 169 @tex |
| 170 $$ | |
| 171 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0 | |
| 172 $$ | |
| 173 @end tex | |
| 6700 | 174 @ifnottex |
| 3294 | 175 |
| 176 @example | |
| 177 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0 | |
| 178 @end example | |
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179 |
| 6700 | 180 @end ifnottex |
| 3294 | 181 |
| 182 @noindent | |
| 6700 | 183 where |
| 184 @tex | |
| 185 $\dot{x} = {dx \over dt}$ | |
| 186 @end tex | |
| 187 @ifnottex | |
| 188 @math{x-dot} | |
| 189 @end ifnottex | |
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190 is the derivative of @math{x}. The equation is solved using |
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191 @nospell{Petzold's} DAE solver @sc{daspk}. |
| 4115 | 192 |
| 193 @DOCSTRING(daspk) | |
| 194 | |
| 195 @DOCSTRING(daspk_options) | |
| 3294 | 196 |
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197 Octave also includes @sc{dassl}, an earlier version of @sc{daspk}, |
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198 and @sc{dasrt}, which can be used to solve DAEs with constraints |
| 4115 | 199 (stopping conditions). |
| 3294 | 200 |
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201 @DOCSTRING(dassl) |
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202 |
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203 @DOCSTRING(dassl_options) |
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204 |
| 4115 | 205 @DOCSTRING(dasrt) |
| 206 | |
| 207 @DOCSTRING(dasrt_options) | |
| 3294 | 208 |
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209 See @nospell{K. E. Brenan}, et al., @cite{Numerical Solution of Initial-Value |
| 3294 | 210 Problems in Differential-Algebraic Equations}, North-Holland (1989) for |
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Grammarcheck .txi documentation files.
Rik <octave@nomad.inbox5.com>
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211 more information about the implementation of @sc{dassl}. |