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  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. There are five different integrate functions which have different
  37         strategies on when to call the observer function during integration. All
  38         of the integrate functions except <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
  39         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
  40         Stepper</a> , <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
  41         Stepper</a> , <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
  42         Output Stepper</a>. Depending on the abilities of the stepper, the integrate
  43         functions make use of step-size control or dense output.
  44       </p>
  45 <h5>
  46 <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h0"></a>
  47         <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
  48         observer calls</a>
  49       </h5>
  50 <p>
  51         If observer calls at equidistant time intervals <span class="emphasis"><em>dt</em></span> are
  52         needed, the <code class="computeroutput"><span class="identifier">integrate_const</span></code>
  53         or <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> function
  54         should be used. We start with explaining <code class="computeroutput"><span class="identifier">integrate_const</span></code>:
  55       </p>
  56 <p>
  57         <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span>
  58         <span class="identifier">stepper</span> <span class="special">,</span>
  59         <span class="identifier">system</span> <span class="special">,</span>
  60         <span class="identifier">x0</span> <span class="special">,</span>
  61         <span class="identifier">t0</span> <span class="special">,</span>
  62         <span class="identifier">t1</span> <span class="special">,</span>
  63         <span class="identifier">dt</span> <span class="special">)</span></code>
  64       </p>
  65 <p>
  66         <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span>
  67         <span class="identifier">stepper</span> <span class="special">,</span>
  68         <span class="identifier">system</span> <span class="special">,</span>
  69         <span class="identifier">x0</span> <span class="special">,</span>
  70         <span class="identifier">t0</span> <span class="special">,</span>
  71         <span class="identifier">t1</span> <span class="special">,</span>
  72         <span class="identifier">dt</span> <span class="special">,</span>
  73         <span class="identifier">observer</span> <span class="special">)</span></code>
  74       </p>
  75 <p>
  76         These integrate the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
  77         with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
  78         Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and
  79         <code class="computeroutput"><span class="identifier">x0</span></code> and ends at some <span class="emphasis"><em>t'
  80         = 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> -
  81         dt &lt; t' &lt;= t<sub>1</sub></em></span>. <code class="computeroutput"><span class="identifier">x0</span></code>
  82         is changed to the approximative solution <span class="emphasis"><em>x(t')</em></span> at the
  83         end of integration. If provided, the <code class="computeroutput"><span class="identifier">observer</span></code>
  84         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>,
  85         <span class="emphasis"><em>t<sub>0</sub> + 2dt</em></span>, ... ,<span class="emphasis"><em>t'</em></span>. <code class="computeroutput"><span class="identifier">integrate_const</span></code> returns the number of steps
  86         performed during the integration. Note that if you are using a simple <a class="link" href="../concepts/stepper.html" title="Stepper">Stepper</a> or <a class="link" href="../concepts/error_stepper.html" title="Error Stepper">Error
  87         Stepper</a> and want to make exactly <code class="computeroutput"><span class="identifier">n</span></code>
  88         steps you should prefer the <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code>
  89         function below.
  90       </p>
  91 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  92 <li class="listitem">
  93             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>
  94             then <code class="computeroutput"><span class="identifier">dt</span></code> is also the step
  95             size used for integration and the observer is called just after every
  96             step.
  97           </li>
  98 <li class="listitem">
  99             If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 100             Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 101             is the initial step size. The actual step size will change due to error
 102             control during time evolution. However, if an observer is provided the
 103             step size will be adjusted such that the algorithm always calculates
 104             <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n dt</em></span> and calls
 105             the observer at that point. Note that the use of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 106             Stepper</a> is reasonable here only if <code class="computeroutput"><span class="identifier">dt</span></code>
 107             is considerably larger than typical step sizes used by the stepper.
 108           </li>
 109 <li class="listitem">
 110             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
 111             Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 112             is the initial step size. The actual step size will be adjusted during
 113             integration due to error control. If an observer is provided dense output
 114             is used to calculate <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n
 115             dt</em></span>.
 116           </li>
 117 </ul></div>
 118 <h5>
 119 <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h1"></a>
 120         <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
 121         a given number of steps</a>
 122       </h5>
 123 <p>
 124         This function is very similar to <code class="computeroutput"><span class="identifier">integrate_const</span></code>
 125         above. The only difference is that it does not take the end time as parameter,
 126         but rather the number of steps. The integration is then performed until the
 127         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>.
 128       </p>
 129 <p>
 130         <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span>
 131         <span class="identifier">stepper</span> <span class="special">,</span>
 132         <span class="identifier">system</span> <span class="special">,</span>
 133         <span class="identifier">x0</span> <span class="special">,</span>
 134         <span class="identifier">t0</span> <span class="special">,</span>
 135         <span class="identifier">dt</span> <span class="special">,</span>
 136         <span class="identifier">n</span> <span class="special">)</span></code>
 137       </p>
 138 <p>
 139         <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span>
 140         <span class="identifier">stepper</span> <span class="special">,</span>
 141         <span class="identifier">system</span> <span class="special">,</span>
 142         <span class="identifier">x0</span> <span class="special">,</span>
 143         <span class="identifier">t0</span> <span class="special">,</span>
 144         <span class="identifier">dt</span> <span class="special">,</span>
 145         <span class="identifier">n</span> <span class="special">,</span> <span class="identifier">observer</span> <span class="special">)</span></code>
 146       </p>
 147 <p>
 148         Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
 149         with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>
 150         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,
 151         <code class="computeroutput"><span class="identifier">observer</span></code> is called after
 152         every step and at the beginning with <code class="computeroutput"><span class="identifier">t0</span></code>,
 153         similar as above. The approximate result for <span class="emphasis"><em>x( t<sub>0</sub> + n dt )</em></span>
 154         is stored in <code class="computeroutput"><span class="identifier">x0</span></code>. This function
 155         returns the end 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>.
 156       </p>
 157 <h5>
 158 <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h2"></a>
 159         <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
 160         calls at each step</a>
 161       </h5>
 162 <p>
 163         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.
 164         Note that in the case of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 165         Stepper</a> or <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
 166         Output Stepper</a> this leads to non-equidistant observer calls as the
 167         step size changes.
 168       </p>
 169 <p>
 170         <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span>
 171         <span class="identifier">stepper</span> <span class="special">,</span>
 172         <span class="identifier">system</span> <span class="special">,</span>
 173         <span class="identifier">x0</span> <span class="special">,</span>
 174         <span class="identifier">t0</span> <span class="special">,</span>
 175         <span class="identifier">t1</span> <span class="special">,</span>
 176         <span class="identifier">dt</span> <span class="special">)</span></code>
 177       </p>
 178 <p>
 179         <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span>
 180         <span class="identifier">stepper</span> <span class="special">,</span>
 181         <span class="identifier">system</span> <span class="special">,</span>
 182         <span class="identifier">x0</span> <span class="special">,</span>
 183         <span class="identifier">t0</span> <span class="special">,</span>
 184         <span class="identifier">t1</span> <span class="special">,</span>
 185         <span class="identifier">dt</span> <span class="special">,</span>
 186         <span class="identifier">observer</span> <span class="special">)</span></code>
 187       </p>
 188 <p>
 189         Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
 190         with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
 191         Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and
 192         <code class="computeroutput"><span class="identifier">x0</span></code> and ends at <span class="emphasis"><em>t<sub>1</sub></em></span>.
 193         <code class="computeroutput"><span class="identifier">x0</span></code> is changed to the approximative
 194         solution <span class="emphasis"><em>x(t<sub>1</sub>)</em></span> at the end of integration. If provided,
 195         the <code class="computeroutput"><span class="identifier">observer</span></code> is called after
 196         each step (and before the first step at <code class="computeroutput"><span class="identifier">t0</span></code>).
 197         <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> returns
 198         the number of steps performed during the integration.
 199       </p>
 200 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 201 <li class="listitem">
 202             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>
 203             then <code class="computeroutput"><span class="identifier">dt</span></code> is the step size
 204             used for integration and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
 205             behaves like <code class="computeroutput"><span class="identifier">integrate_const</span></code>
 206             except that for the last step the step size is reduced to ensure we end
 207             exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. If provided,
 208             the observer is called at each step.
 209           </li>
 210 <li class="listitem">
 211             If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 212             Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 213             is the initial step size. The actual step size is changed according to
 214             error control of the stepper. For the last step, the step size will be
 215             reduced to ensure we end exactly at <code class="computeroutput"><span class="identifier">t1</span></code>.
 216             If provided, the observer is called after each time step (and before
 217             the first step at <code class="computeroutput"><span class="identifier">t0</span></code>).
 218           </li>
 219 <li class="listitem">
 220             If stepper is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense
 221             Output Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 222             is the initial step size and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
 223             behaves just like for <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 224             Stepper</a> above. No dense output is used.
 225           </li>
 226 </ul></div>
 227 <h5>
 228 <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h3"></a>
 229         <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
 230         calls at given time points</a>
 231       </h5>
 232 <p>
 233         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.
 234         The times for observer calls are provided as a sequence of time values. The
 235         sequence is either defined via two iterators pointing to begin and end of
 236         the sequence or in terms of a <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>
 237         object.
 238       </p>
 239 <p>
 240         <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span>
 241         <span class="identifier">stepper</span> <span class="special">,</span>
 242         <span class="identifier">system</span> <span class="special">,</span>
 243         <span class="identifier">x0</span> <span class="special">,</span>
 244         <span class="identifier">times_start</span> <span class="special">,</span>
 245         <span class="identifier">times_end</span> <span class="special">,</span>
 246         <span class="identifier">dt</span> <span class="special">,</span>
 247         <span class="identifier">observer</span> <span class="special">)</span></code>
 248       </p>
 249 <p>
 250         <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span>
 251         <span class="identifier">stepper</span> <span class="special">,</span>
 252         <span class="identifier">system</span> <span class="special">,</span>
 253         <span class="identifier">x0</span> <span class="special">,</span>
 254         <span class="identifier">time_range</span> <span class="special">,</span>
 255         <span class="identifier">dt</span> <span class="special">,</span>
 256         <span class="identifier">observer</span> <span class="special">)</span></code>
 257       </p>
 258 <p>
 259         Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code>
 260         with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>.
 261         Integration starts at <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span></code>
 262         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>.
 263         <code class="computeroutput"><span class="identifier">x0</span></code> contains the approximate
 264         solution at the end point of integration. This function requires an observer
 265         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>
 266         until <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">==</span>
 267         <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>
 268         <code class="computeroutput"><span class="identifier">time_range</span></code> the function behaves
 269         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>
 270         <span class="special">)</span></code> and <code class="computeroutput"><span class="identifier">times_end</span>
 271         <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">end</span><span class="special">(</span>
 272         <span class="identifier">time_range</span> <span class="special">)</span></code>.
 273         <code class="computeroutput"><span class="identifier">integrate_times</span></code> returns the
 274         number of steps performed during the integration.
 275       </p>
 276 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 277 <li class="listitem">
 278             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>
 279             <code class="computeroutput"><span class="identifier">dt</span></code> is the step size used
 280             for integration. However, whenever a time point from the sequence is
 281             approached the step size <code class="computeroutput"><span class="identifier">dt</span></code>
 282             will be reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> exactly
 283             at the time point.
 284           </li>
 285 <li class="listitem">
 286             If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled
 287             Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 288             is the initial step size. The actual step size is adjusted during integration
 289             according to error control. However, if a time point from the sequence
 290             is approached the step size is reduced to obtain the state <span class="emphasis"><em>x(t)</em></span>
 291             exactly at the time point.
 292           </li>
 293 <li class="listitem">
 294             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
 295             Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code>
 296             is the initial step size. The actual step size is adjusted during integration
 297             according to error control. Dense output is used to obtain the states
 298             <span class="emphasis"><em>x(t)</em></span> at the time points from the sequence.
 299           </li>
 300 </ul></div>
 301 <h5>
 302 <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h4"></a>
 303         <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
 304         integrate function</a>
 305       </h5>
 306 <p>
 307         Additionally to the sophisticated integrate function above odeint also provides
 308         a simple <code class="computeroutput"><span class="identifier">integrate</span></code> routine
 309         which uses a dense output stepper based on <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code>
 310         with standard error bounds <span class="emphasis"><em>10<sup>-6</sup></em></span> for the steps.
 311       </p>
 312 <p>
 313         <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span>
 314         <span class="identifier">system</span> <span class="special">,</span>
 315         <span class="identifier">x0</span> <span class="special">,</span>
 316         <span class="identifier">t0</span> <span class="special">,</span>
 317         <span class="identifier">t1</span> <span class="special">,</span>
 318         <span class="identifier">dt</span> <span class="special">)</span></code>
 319       </p>
 320 <p>
 321         <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span>
 322         <span class="identifier">system</span> <span class="special">,</span>
 323         <span class="identifier">x0</span> <span class="special">,</span>
 324         <span class="identifier">t0</span> <span class="special">,</span>
 325         <span class="identifier">t1</span> <span class="special">,</span>
 326         <span class="identifier">dt</span> <span class="special">,</span>
 327         <span class="identifier">observer</span> <span class="special">)</span></code>
 328       </p>
 329 <p>
 330         This function behaves exactly like <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code>
 331         above but no stepper has to be provided. It also returns the number of steps
 332         performed during the integration.
 333       </p>
 334 </div>
 335 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 336 <td align="left"></td>
 337 <td align="right"><div class="copyright-footer">Copyright &#169; 2009-2012 Karsten
 338       Ahnert and Mario Mulansky<p>
 339         Distributed under the Boost Software License, Version 1.0. (See accompanying
 340         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>)
 341       </p>
 342 </div></td>
 343 </tr></table>
 344 <hr>
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