1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Implicit System</title> 5<link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint"> 8<link rel="up" href="../concepts.html" title="Concepts"> 9<link rel="prev" href="simple_symplectic_system.html" title="Simple Symplectic System"> 10<link rel="next" href="stepper.html" title="Stepper"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../logo.jpg"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="simple_symplectic_system.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="stepper.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="boost_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit 28 System</a> 29</h3></div></div></div> 30<h5> 31<a name="boost_numeric_odeint.concepts.implicit_system.h0"></a> 32 <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.description"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.description">Description</a> 33 </h5> 34<p> 35 This concept describes how to define a ODE that can be solved by an implicit 36 routine. Implicit routines need not only the function <span class="emphasis"><em>f(x,t)</em></span> 37 but also the Jacobian <span class="emphasis"><em>df/dx = A(x,t)</em></span>. <span class="emphasis"><em>A</em></span> 38 is a matrix and implicit routines need to solve the linear problem <span class="emphasis"><em>Ax 39 = b</em></span>. In odeint this is implemented with use of <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>, 40 therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span> 41 and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>. 42 </p> 43<h5> 44<a name="boost_numeric_odeint.concepts.implicit_system.h1"></a> 45 <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.notation"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.notation">Notation</a> 46 </h5> 47<div class="variablelist"> 48<p class="title"><b></b></p> 49<dl class="variablelist"> 50<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt> 51<dd><p> 52 A type that is a model of <code class="computeroutput"><span class="identifier">Implicit</span> 53 <span class="identifier">System</span></code> 54 </p></dd> 55<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt> 56<dd><p> 57 A type representing the time of the ODE 58 </p></dd> 59<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt> 60<dd><p> 61 An object of type <code class="computeroutput"><span class="identifier">System</span></code> 62 </p></dd> 63<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt> 64<dd><p> 65 Object of type ublas::vector 66 </p></dd> 67<dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt> 68<dd><p> 69 Object of type ublas::vector 70 </p></dd> 71<dt><span class="term"><code class="computeroutput"><span class="identifier">jacobi</span></code></span></dt> 72<dd><p> 73 Object of type ublas::matrix 74 </p></dd> 75<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt> 76<dd><p> 77 Object of type <code class="computeroutput"><span class="identifier">Time</span></code> 78 </p></dd> 79</dl> 80</div> 81<h5> 82<a name="boost_numeric_odeint.concepts.implicit_system.h2"></a> 83 <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.valid_expressions"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.valid_expressions">Valid 84 Expressions</a> 85 </h5> 86<div class="informaltable"><table class="table"> 87<colgroup> 88<col> 89<col> 90<col> 91<col> 92</colgroup> 93<thead><tr> 94<th> 95 <p> 96 Name 97 </p> 98 </th> 99<th> 100 <p> 101 Expression 102 </p> 103 </th> 104<th> 105 <p> 106 Type 107 </p> 108 </th> 109<th> 110 <p> 111 Semantics 112 </p> 113 </th> 114</tr></thead> 115<tbody> 116<tr> 117<td> 118 <p> 119 Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span> 120 </p> 121 </td> 122<td> 123 <p> 124 <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span> 125 <span class="identifier">x</span> <span class="special">,</span> 126 <span class="identifier">dxdt</span> <span class="special">,</span> 127 <span class="identifier">t</span> <span class="special">)</span></code> 128 </p> 129 </td> 130<td> 131 <p> 132 <code class="computeroutput"><span class="keyword">void</span></code> 133 </p> 134 </td> 135<td> 136 <p> 137 Calculates <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span><span class="identifier">t</span><span class="special">)</span></code>, 138 the result is stored into dxdt 139 </p> 140 </td> 141</tr> 142<tr> 143<td> 144 <p> 145 Calculate <span class="emphasis"><em>A := df/dx (x,t)</em></span> 146 </p> 147 </td> 148<td> 149 <p> 150 <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span> 151 <span class="identifier">x</span> <span class="special">,</span> 152 <span class="identifier">jacobi</span> <span class="special">,</span> 153 <span class="identifier">t</span> <span class="special">)</span></code> 154 </p> 155 </td> 156<td> 157 <p> 158 <code class="computeroutput"><span class="keyword">void</span></code> 159 </p> 160 </td> 161<td> 162 <p> 163 Calculates the Jacobian of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>t</em></span>, 164 the result is stored into <code class="computeroutput"><span class="identifier">jacobi</span></code> 165 </p> 166 </td> 167</tr> 168</tbody> 169</table></div> 170</div> 171<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 172<td align="left"></td> 173<td align="right"><div class="copyright-footer">Copyright © 2009-2015 Karsten Ahnert and Mario Mulansky<p> 174 Distributed under the Boost Software License, Version 1.0. (See accompanying 175 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 176 </p> 177</div></td> 178</tr></table> 179<hr> 180<div class="spirit-nav"> 181<a accesskey="p" href="simple_symplectic_system.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="stepper.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 182</div> 183</body> 184</html> 185
