annotate doc/interpreter/diffeq.txi @ 8828:8463d1a2e544

Doc fixes. * 2]$$. => 2].$$ * @var{extrapval} => @var{extrapval}. * call helloworld.oct => called @file{helloworld.oct} * @itemize => @table @code * shows. => shows: * save => @code{save} * @ref{Breakpoints} => @pxref{Breakpoints} * add @noindent following example * which is computed => and compute it * clarify wording * remove comma * good => well * set => number * by writing => with the command * has the option of directly calling => can call * [-like-] {+of the right size,+} * solvers => routines * handle => test for * add introductory section * add following * {+the+} [0..bitmax] => [0,bitmax] * of the => with * number => value * add usual * Besides when doing comparisons, logical => Logical {+also+} * array comparison => array, comparisons * param => parameter * works very similar => is similar * strings, => strings * most simple => simplest * easier => more easily * like => as * called => called, * clarify wording * you should simply type => use * clarify wording * means => way * equally => also * [-way much-] {+way+} * add with mean value parameter given by the first argument, @var{l} * add Functions described as @dfn{mapping functions} apply the given operation to each element when given a matrix argument. * in this brief introduction => here * It is worth noticing => Note * add following * means => ways
author Brian Gough <bjg@network-theory.co.uk>
date Fri, 20 Feb 2009 11:17:01 -0500
parents 6f2d95255911
children eb63fbe60fab
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1 @c Copyright (C) 1996, 1997, 2007 John W. Eaton
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2 @c
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3 @c This file is part of Octave.
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4 @c
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5 @c Octave is free software; you can redistribute it and/or modify it
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6 @c under the terms of the GNU General Public License as published by the
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7 @c Free Software Foundation; either version 3 of the License, or (at
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8 @c your option) any later version.
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9 @c
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10 @c Octave is distributed in the hope that it will be useful, but WITHOUT
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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13 @c for more details.
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14 @c
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15 @c You should have received a copy of the GNU General Public License
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16 @c along with Octave; see the file COPYING. If not, see
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17 @c <http://www.gnu.org/licenses/>.
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18
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19 @node Differential Equations
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20 @chapter Differential Equations
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21
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22 Octave has built-in functions for solving ordinary differential equations,
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23 and differential-algebraic equations.
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24 All solvers are based on reliable ODE routines written in Fortran.
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25
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26 @menu
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27 * Ordinary Differential Equations::
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28 * Differential-Algebraic Equations::
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29 @end menu
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30
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31 @cindex Differential Equations
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32 @cindex ODE
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33 @cindex DAE
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34
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35 @node Ordinary Differential Equations
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36 @section Ordinary Differential Equations
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37
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38 The function @code{lsode} can be used to solve ODEs of the form
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39 @iftex
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40 @tex
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41 $$
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42 {dx\over dt} = f (x, t)
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43 $$
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44 @end tex
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45 @end iftex
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46 @ifinfo
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47
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48 @example
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49 dx
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50 -- = f (x, t)
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51 dt
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52 @end example
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53 @end ifinfo
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54
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55 @noindent
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56 using Hindmarsh's ODE solver @sc{Lsode}.
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57
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58
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59
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60 @DOCSTRING(lsode)
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61
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62 @DOCSTRING(lsode_options)
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63
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64 Here is an example of solving a set of three differential equations using
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65 @code{lsode}. Given the function
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66
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67 @cindex oregonator
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68
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69 @example
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70 @group
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71 function xdot = f (x, t)
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72
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73 xdot = zeros (3,1);
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74
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75 xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \
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76 - 8.375e-06*x(1)^2);
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77 xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
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78 xdot(3) = 0.161*(x(1) - x(3));
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79
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80 endfunction
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81 @end group
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82 @end example
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83
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84 @noindent
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85 and the initial condition @code{x0 = [ 4; 1.1; 4 ]}, the set of
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86 equations can be integrated using the command
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87
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88 @example
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89 @group
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90 t = linspace (0, 500, 1000);
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91
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92 y = lsode ("f", x0, t);
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93 @end group
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94 @end example
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95
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96 If you try this, you will see that the value of the result changes
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97 dramatically between @var{t} = 0 and 5, and again around @var{t} = 305.
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98 A more efficient set of output points might be
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99
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100 @example
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101 @group
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102 t = [0, logspace (-1, log10(303), 150), \
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103 logspace (log10(304), log10(500), 150)];
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104 @end group
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105 @end example
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106
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107 See Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE
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108 Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for
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109 more information about the inner workings of @code{lsode}.
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110
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111 @node Differential-Algebraic Equations
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112 @section Differential-Algebraic Equations
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113
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114 The function @code{daspk} can be used to solve DAEs of the form
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115 @iftex
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116 @tex
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117 $$
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118 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0
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119 $$
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120 @end tex
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121 @end iftex
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122 @ifnottex
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123
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124 @example
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125 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0
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126 @end example
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127 @end ifnottex
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128
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129 @noindent
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130 where
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131 @iftex
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132 @tex
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133 $\dot{x} = {dx \over dt}$
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134 @end tex
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135 @end iftex
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136 @ifnottex
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137 @math{x-dot}
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138 @end ifnottex
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139 is the derivative of @math{x}. The equation is solved using Petzold's
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140 DAE solver @sc{Daspk}.
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141
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142 @DOCSTRING(daspk)
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143
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144 @DOCSTRING(daspk_options)
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145
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146 Octave also includes @sc{Dassl}, an earlier version of @var{Daspk},
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147 and @var{dasrt}, which can be used to solve DAEs with constraints
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148 (stopping conditions).
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149
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150 @DOCSTRING(dassl)
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151
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152 @DOCSTRING(dassl_options)
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153
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154 @DOCSTRING(dasrt)
fc2048d4cd21 [project @ 2002-10-22 21:28:42 by jwe]
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155
fc2048d4cd21 [project @ 2002-10-22 21:28:42 by jwe]
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156 @DOCSTRING(dasrt_options)
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bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe]
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157
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe]
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158 See K. E. Brenan, et al., @cite{Numerical Solution of Initial-Value
bfe1573bd2ae [project @ 1999-10-19 10:06:07 by jwe]
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159 Problems in Differential-Algebraic Equations}, North-Holland (1989) for
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160 more information about the implementation of @sc{Dassl}.