1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Integrate functions</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="../odeint_in_detail.html" title="odeint in detail"> 9<link rel="prev" href="generation_functions.html" title="Generation functions"> 10<link rel="next" href="iterators_and_ranges.html" title="Iterators and Ranges"> 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="generation_functions.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.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="iterators_and_ranges.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.odeint_in_detail.integrate_functions"></a><a class="link" href="integrate_functions.html" title="Integrate functions">Integrate 28 functions</a> 29</h3></div></div></div> 30<p> 31 Integrate functions perform the time evolution of a given ODE from some starting 32 time <span class="emphasis"><em>t<sub>0</sub></em></span> to a given end time <span class="emphasis"><em>t<sub>1</sub></em></span> 33 and starting at state <span class="emphasis"><em>x<sub>0</sub></em></span> by subsequent calls of a given 34 stepper's <code class="computeroutput"><span class="identifier">do_step</span></code> function. 35 Additionally, the user can provide an __observer to analyze the state during 36 time evolution, and a <code class="computeroutput"><a class="link" href="../../boost/numeric/odeint/max_step_checker.html" title="Class max_step_checker">max_step_checker</a></code> to throw an 37 exception if too many steps are taken between observer calls (i.e. too small 38 step size). There are five different integrate functions which have different 39 strategies on when to call the observer function during integration. All 40 of the integrate functions except <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> 41 can be called with any stepper following one of the stepper concepts: <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> , <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error 42 Stepper</a> , <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 43 Stepper</a> , <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense 44 Output Stepper</a>. Depending on the abilities of the stepper, the integrate 45 functions make use of step-size control or dense output. 46 </p> 47<h5> 48<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h0"></a> 49 <span class="phrase"><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.equidistant_observer_calls">Equidistant 50 observer calls</a> 51 </h5> 52<p> 53 If observer calls at equidistant time intervals <span class="emphasis"><em>dt</em></span> are 54 needed, the <code class="computeroutput"><span class="identifier">integrate_const</span></code> 55 or <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> function 56 should be used. We start with explaining <code class="computeroutput"><span class="identifier">integrate_const</span></code>: 57 </p> 58<p> 59 <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span> 60 <span class="identifier">stepper</span> <span class="special">,</span> 61 <span class="identifier">system</span> <span class="special">,</span> 62 <span class="identifier">x0</span> <span class="special">,</span> 63 <span class="identifier">t0</span> <span class="special">,</span> 64 <span class="identifier">t1</span> <span class="special">,</span> 65 <span class="identifier">dt</span> <span class="special">)</span></code> 66 </p> 67<p> 68 <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span> 69 <span class="identifier">stepper</span> <span class="special">,</span> 70 <span class="identifier">system</span> <span class="special">,</span> 71 <span class="identifier">x0</span> <span class="special">,</span> 72 <span class="identifier">t0</span> <span class="special">,</span> 73 <span class="identifier">t1</span> <span class="special">,</span> 74 <span class="identifier">dt</span> <span class="special">,</span> 75 <span class="identifier">observer</span> <span class="special">)</span></code> 76 </p> 77<p> 78 <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span> 79 <span class="identifier">stepper</span> <span class="special">,</span> 80 <span class="identifier">system</span> <span class="special">,</span> 81 <span class="identifier">x0</span> <span class="special">,</span> 82 <span class="identifier">t0</span> <span class="special">,</span> 83 <span class="identifier">t1</span> <span class="special">,</span> 84 <span class="identifier">dt</span> <span class="special">,</span> 85 <span class="identifier">observer</span> <span class="special">,</span> 86 <span class="identifier">max_step_checker</span> <span class="special">)</span></code> 87 </p> 88<p> 89 These integrate the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> 90 with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. 91 Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and 92 <code class="computeroutput"><span class="identifier">x0</span></code> and ends at some <span class="emphasis"><em>t' 93 = t<sub>0</sub> + n dt</em></span> with <span class="emphasis"><em>n</em></span> such that <span class="emphasis"><em>t<sub>1</sub> - 94 dt < t' <= t<sub>1</sub></em></span>. <code class="computeroutput"><span class="identifier">x0</span></code> 95 is changed to the approximative solution <span class="emphasis"><em>x(t')</em></span> at the 96 end of integration. If provided, the <code class="computeroutput"><span class="identifier">observer</span></code> 97 is invoked at times <span class="emphasis"><em>t<sub>0</sub></em></span>, <span class="emphasis"><em>t<sub>0</sub> + dt</em></span>, 98 <span class="emphasis"><em>t<sub>0</sub> + 2dt</em></span>, ... ,<span class="emphasis"><em>t'</em></span>. If provided, 99 the <code class="computeroutput"><span class="identifier">max_step_checker</span></code> counts 100 the number of steps between observer calls and throws a <code class="computeroutput"><span class="identifier">no_progress_error</span></code> 101 this exceeds some limit (default: 500). <code class="computeroutput"><span class="identifier">integrate_const</span></code> 102 returns the number of steps performed during the integration. Note that if 103 you are using a simple <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> 104 or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a> 105 and want to make exactly <code class="computeroutput"><span class="identifier">n</span></code> 106 steps you should prefer the <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> 107 function below. 108 </p> 109<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 110<li class="listitem"> 111 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a> 112 then <code class="computeroutput"><span class="identifier">dt</span></code> is also the step 113 size used for integration and the observer is called just after every 114 step. 115 </li> 116<li class="listitem"> 117 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 118 Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 119 is the initial step size. The actual step size will change due to error 120 control during time evolution. However, if an observer is provided the 121 step size will be adjusted such that the algorithm always calculates 122 <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n dt</em></span> and calls 123 the observer at that point. Note that the use of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 124 Stepper</a> is reasonable here only if <code class="computeroutput"><span class="identifier">dt</span></code> 125 is considerably larger than typical step sizes used by the stepper. 126 </li> 127<li class="listitem"> 128 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense Output 129 Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 130 is the initial step size. The actual step size will be adjusted during 131 integration due to error control. If an observer is provided dense output 132 is used to calculate <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n 133 dt</em></span>. 134 </li> 135</ul></div> 136<h5> 137<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h1"></a> 138 <span class="phrase"><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.integrate_a_given_number_of_steps">Integrate 139 a given number of steps</a> 140 </h5> 141<p> 142 This function is very similar to <code class="computeroutput"><span class="identifier">integrate_const</span></code> 143 above. The only difference is that it does not take the end time as parameter, 144 but rather the number of steps. The integration is then performed until the 145 time <code class="computeroutput"><span class="identifier">t0</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>. 146 </p> 147<p> 148 <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span> 149 <span class="identifier">stepper</span> <span class="special">,</span> 150 <span class="identifier">system</span> <span class="special">,</span> 151 <span class="identifier">x0</span> <span class="special">,</span> 152 <span class="identifier">t0</span> <span class="special">,</span> 153 <span class="identifier">dt</span> <span class="special">,</span> 154 <span class="identifier">n</span> <span class="special">)</span></code> 155 </p> 156<p> 157 <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span> 158 <span class="identifier">stepper</span> <span class="special">,</span> 159 <span class="identifier">system</span> <span class="special">,</span> 160 <span class="identifier">x0</span> <span class="special">,</span> 161 <span class="identifier">t0</span> <span class="special">,</span> 162 <span class="identifier">dt</span> <span class="special">,</span> 163 <span class="identifier">n</span> <span class="special">,</span> <span class="identifier">observer</span> <span class="special">)</span></code> 164 </p> 165<p> 166 <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span> 167 <span class="identifier">stepper</span> <span class="special">,</span> 168 <span class="identifier">system</span> <span class="special">,</span> 169 <span class="identifier">x0</span> <span class="special">,</span> 170 <span class="identifier">t0</span> <span class="special">,</span> 171 <span class="identifier">dt</span> <span class="special">,</span> 172 <span class="identifier">n</span> <span class="special">,</span> <span class="identifier">observer</span> <span class="special">,</span> <span class="identifier">max_step_checker</span> <span class="special">)</span></code> 173 </p> 174<p> 175 Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> 176 with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code> 177 starting at <span class="emphasis"><em>x<sub>0</sub></em></span> and <span class="emphasis"><em>t<sub>0</sub></em></span>. If provided, 178 <code class="computeroutput"><span class="identifier">observer</span></code> is called after 179 every step and at the beginning with <code class="computeroutput"><span class="identifier">t0</span></code>, 180 similar as above. Again, providing a <code class="computeroutput"><span class="identifier">max_step_checker</span></code> 181 will throw a <code class="computeroutput"><span class="identifier">no_progress_error</span></code> 182 if too many steps are performed between observer calls. The approximate result 183 for <span class="emphasis"><em>x( t<sub>0</sub> + n dt )</em></span> is stored in <code class="computeroutput"><span class="identifier">x0</span></code>. 184 This function returns the end time <code class="computeroutput"><span class="identifier">t0</span> 185 <span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">dt</span></code>. 186 </p> 187<h5> 188<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h2"></a> 189 <span class="phrase"><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_each_step">Observer 190 calls at each step</a> 191 </h5> 192<p> 193 If the observer should be called at each time step then the <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> function should be used. 194 Note that in the case of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 195 Stepper</a> or <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense 196 Output Stepper</a> this leads to non-equidistant observer calls as the 197 step size changes. 198 </p> 199<p> 200 <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span> 201 <span class="identifier">stepper</span> <span class="special">,</span> 202 <span class="identifier">system</span> <span class="special">,</span> 203 <span class="identifier">x0</span> <span class="special">,</span> 204 <span class="identifier">t0</span> <span class="special">,</span> 205 <span class="identifier">t1</span> <span class="special">,</span> 206 <span class="identifier">dt</span> <span class="special">)</span></code> 207 </p> 208<p> 209 <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span> 210 <span class="identifier">stepper</span> <span class="special">,</span> 211 <span class="identifier">system</span> <span class="special">,</span> 212 <span class="identifier">x0</span> <span class="special">,</span> 213 <span class="identifier">t0</span> <span class="special">,</span> 214 <span class="identifier">t1</span> <span class="special">,</span> 215 <span class="identifier">dt</span> <span class="special">,</span> 216 <span class="identifier">observer</span> <span class="special">)</span></code> 217 </p> 218<p> 219 Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> 220 with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. 221 Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and 222 <code class="computeroutput"><span class="identifier">x0</span></code> and ends at <span class="emphasis"><em>t<sub>1</sub></em></span>. 223 <code class="computeroutput"><span class="identifier">x0</span></code> is changed to the approximative 224 solution <span class="emphasis"><em>x(t<sub>1</sub>)</em></span> at the end of integration. If provided, 225 the <code class="computeroutput"><span class="identifier">observer</span></code> is called after 226 each step (and before the first step at <code class="computeroutput"><span class="identifier">t0</span></code>). 227 <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> returns 228 the number of steps performed during the integration. 229 </p> 230<div class="note"><table border="0" summary="Note"> 231<tr> 232<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 233<th align="left">Note</th> 234</tr> 235<tr><td align="left" valign="top"><p> 236 <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> by design 237 performs an observer call after each time step. Hence there is no need 238 for a <code class="computeroutput"><a class="link" href="../../boost/numeric/odeint/max_step_checker.html" title="Class max_step_checker">max_step_checker</a></code> as only 239 exactly one step is ever performed between observer calls. 240 </p></td></tr> 241</table></div> 242<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 243<li class="listitem"> 244 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a> 245 then <code class="computeroutput"><span class="identifier">dt</span></code> is the step size 246 used for integration and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> 247 behaves like <code class="computeroutput"><span class="identifier">integrate_const</span></code> 248 except that for the last step the step size is reduced to ensure we end 249 exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. If provided, 250 the observer is called at each step. 251 </li> 252<li class="listitem"> 253 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 254 Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 255 is the initial step size. The actual step size is changed according to 256 error control of the stepper. For the last step, the step size will be 257 reduced to ensure we end exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. 258 If provided, the observer is called after each time step (and before 259 the first step at <code class="computeroutput"><span class="identifier">t0</span></code>). 260 </li> 261<li class="listitem"> 262 If stepper is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense 263 Output Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 264 is the initial step size and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> 265 behaves just like for <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 266 Stepper</a> above. No dense output is used. 267 </li> 268</ul></div> 269<h5> 270<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h3"></a> 271 <span class="phrase"><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.observer_calls_at_given_time_points">Observer 272 calls at given time points</a> 273 </h5> 274<p> 275 If the observer should be called at some user given time points the <code class="computeroutput"><span class="identifier">integrate_times</span></code> function should be used. 276 The times for observer calls are provided as a sequence of time values. The 277 sequence is either defined via two iterators pointing to begin and end of 278 the sequence or in terms of a <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a> 279 object. 280 </p> 281<p> 282 <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span> 283 <span class="identifier">stepper</span> <span class="special">,</span> 284 <span class="identifier">system</span> <span class="special">,</span> 285 <span class="identifier">x0</span> <span class="special">,</span> 286 <span class="identifier">times_start</span> <span class="special">,</span> 287 <span class="identifier">times_end</span> <span class="special">,</span> 288 <span class="identifier">dt</span> <span class="special">,</span> 289 <span class="identifier">observer</span> <span class="special">)</span></code> 290 </p> 291<p> 292 <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span> 293 <span class="identifier">stepper</span> <span class="special">,</span> 294 <span class="identifier">system</span> <span class="special">,</span> 295 <span class="identifier">x0</span> <span class="special">,</span> 296 <span class="identifier">time_range</span> <span class="special">,</span> 297 <span class="identifier">dt</span> <span class="special">,</span> 298 <span class="identifier">observer</span> <span class="special">)</span></code> 299 </p> 300<p> 301 Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> 302 with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. 303 Integration starts at <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span></code> 304 and ends exactly at <code class="computeroutput"><span class="special">*(</span><span class="identifier">times_end</span><span class="special">-</span><span class="number">1</span><span class="special">)</span></code>. 305 <code class="computeroutput"><span class="identifier">x0</span></code> contains the approximate 306 solution at the end point of integration. This function requires an observer 307 which is invoked at the subsequent times <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span><span class="special">++</span></code> 308 until <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">==</span> 309 <span class="identifier">times_end</span></code>. If called with a <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a> 310 <code class="computeroutput"><span class="identifier">time_range</span></code> the function behaves 311 the same with <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">time_range</span> 312 <span class="special">)</span></code> and <code class="computeroutput"><span class="identifier">times_end</span> 313 <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">end</span><span class="special">(</span> 314 <span class="identifier">time_range</span> <span class="special">)</span></code>. 315 Additionally, a <code class="computeroutput"><a class="link" href="../../boost/numeric/odeint/max_step_checker.html" title="Class max_step_checker">max_step_checker</a></code> can be provided, 316 e.g.: 317 </p> 318<p> 319 <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span> 320 <span class="identifier">stepper</span> <span class="special">,</span> 321 <span class="identifier">system</span> <span class="special">,</span> 322 <span class="identifier">x0</span> <span class="special">,</span> 323 <span class="identifier">times_start</span> <span class="special">,</span> 324 <span class="identifier">times_end</span> <span class="special">,</span> 325 <span class="identifier">dt</span> <span class="special">,</span> 326 <span class="identifier">observer</span> <span class="special">,</span> 327 <span class="identifier">max_step_checker</span><span class="special">)</span></code> 328 </p> 329<p> 330 As above, this will throw a <code class="computeroutput"><span class="identifier">no_progress_error</span></code> 331 if too many steps are performed between observer calls. 332 </p> 333<p> 334 <code class="computeroutput"><span class="identifier">integrate_times</span></code> returns the 335 number of steps performed during the integration. 336 </p> 337<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 338<li class="listitem"> 339 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error Stepper</a> 340 <code class="computeroutput"><span class="identifier">dt</span></code> is the step size used 341 for integration. However, whenever a time point from the sequence is 342 approached the step size <code class="computeroutput"><span class="identifier">dt</span></code> 343 will be reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> exactly 344 at the time point. 345 </li> 346<li class="listitem"> 347 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled 348 Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 349 is the initial step size. The actual step size is adjusted during integration 350 according to error control. However, if a time point from the sequence 351 is approached the step size is reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> 352 exactly at the time point. 353 </li> 354<li class="listitem"> 355 If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense Output 356 Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> 357 is the initial step size. The actual step size is adjusted during integration 358 according to error control. Dense output is used to obtain the states 359 <span class="emphasis"><em>x(t)</em></span> at the time points from the sequence. 360 </li> 361</ul></div> 362<h5> 363<a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h4"></a> 364 <span class="phrase"><a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function"></a></span><a class="link" href="integrate_functions.html#boost_numeric_odeint.odeint_in_detail.integrate_functions.convenience_integrate_function">Convenience 365 integrate function</a> 366 </h5> 367<p> 368 Additionally to the sophisticated integrate function above odeint also provides 369 a simple <code class="computeroutput"><span class="identifier">integrate</span></code> routine 370 which uses a dense output stepper based on <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code> 371 with standard error bounds <span class="emphasis"><em>10<sup>-6</sup></em></span> for the steps. 372 </p> 373<p> 374 <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span> 375 <span class="identifier">system</span> <span class="special">,</span> 376 <span class="identifier">x0</span> <span class="special">,</span> 377 <span class="identifier">t0</span> <span class="special">,</span> 378 <span class="identifier">t1</span> <span class="special">,</span> 379 <span class="identifier">dt</span> <span class="special">)</span></code> 380 </p> 381<p> 382 <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span> 383 <span class="identifier">system</span> <span class="special">,</span> 384 <span class="identifier">x0</span> <span class="special">,</span> 385 <span class="identifier">t0</span> <span class="special">,</span> 386 <span class="identifier">t1</span> <span class="special">,</span> 387 <span class="identifier">dt</span> <span class="special">,</span> 388 <span class="identifier">observer</span> <span class="special">)</span></code> 389 </p> 390<p> 391 This function behaves exactly like <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> 392 above but no stepper has to be provided. It also returns the number of steps 393 performed during the integration. 394 </p> 395</div> 396<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 397<td align="left"></td> 398<td align="right"><div class="copyright-footer">Copyright © 2009-2015 Karsten Ahnert and Mario Mulansky<p> 399 Distributed under the Boost Software License, Version 1.0. (See accompanying 400 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>) 401 </p> 402</div></td> 403</tr></table> 404<hr> 405<div class="spirit-nav"> 406<a accesskey="p" href="generation_functions.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../odeint_in_detail.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="iterators_and_ranges.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 407</div> 408</body> 409</html> 410
