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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions"></a><a class="link" href="integrate_functions.html" title="Integrate functions">Integrate |
| functions</a> |
| </h3></div></div></div> |
| <p> |
| Integrate functions perform the time evolution of a given ODE from some starting |
| 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> |
| and starting at state <span class="emphasis"><em>x<sub>0</sub></em></span> by subsequent calls of a given |
| stepper's <code class="computeroutput"><span class="identifier">do_step</span></code> function. |
| Additionally, the user can provide an __observer to analyze the state during |
| time evolution. There are five different integrate functions which have different |
| strategies on when to call the observer function during integration. All |
| of the integrate functions except <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> |
| 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 |
| Stepper</a> , <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> , <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense |
| Output Stepper</a>. Depending on the abilities of the stepper, the integrate |
| functions make use of step-size control or dense output. |
| </p> |
| <h5> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h0"></a> |
| <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 |
| observer calls</a> |
| </h5> |
| <p> |
| If observer calls at equidistant time intervals <span class="emphasis"><em>dt</em></span> are |
| needed, the <code class="computeroutput"><span class="identifier">integrate_const</span></code> |
| or <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> function |
| should be used. We start with explaining <code class="computeroutput"><span class="identifier">integrate_const</span></code>: |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_const</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| These integrate the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> |
| with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. |
| Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and |
| <code class="computeroutput"><span class="identifier">x0</span></code> and ends at some <span class="emphasis"><em>t' |
| = 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> - |
| dt < t' <= t<sub>1</sub></em></span>. <code class="computeroutput"><span class="identifier">x0</span></code> |
| is changed to the approximative solution <span class="emphasis"><em>x(t')</em></span> at the |
| end of integration. If provided, the <code class="computeroutput"><span class="identifier">observer</span></code> |
| 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>, |
| <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 |
| 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 |
| Stepper</a> and want to make exactly <code class="computeroutput"><span class="identifier">n</span></code> |
| steps you should prefer the <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> |
| function below. |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 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> |
| then <code class="computeroutput"><span class="identifier">dt</span></code> is also the step |
| size used for integration and the observer is called just after every |
| step. |
| </li> |
| <li class="listitem"> |
| If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size. The actual step size will change due to error |
| control during time evolution. However, if an observer is provided the |
| step size will be adjusted such that the algorithm always calculates |
| <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n dt</em></span> and calls |
| the observer at that point. Note that the use of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> is reasonable here only if <code class="computeroutput"><span class="identifier">dt</span></code> |
| is considerably larger than typical step sizes used by the stepper. |
| </li> |
| <li class="listitem"> |
| 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 |
| Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size. The actual step size will be adjusted during |
| integration due to error control. If an observer is provided dense output |
| is used to calculate <span class="emphasis"><em>x(t)</em></span> at <span class="emphasis"><em>t = t<sub>0</sub> + n |
| dt</em></span>. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h1"></a> |
| <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 |
| a given number of steps</a> |
| </h5> |
| <p> |
| This function is very similar to <code class="computeroutput"><span class="identifier">integrate_const</span></code> |
| above. The only difference is that it does not take the end time as parameter, |
| but rather the number of steps. The integration is then performed until the |
| 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>. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">n</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_n_steps</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">n</span> <span class="special">,</span> <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> |
| with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code> |
| 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, |
| <code class="computeroutput"><span class="identifier">observer</span></code> is called after |
| every step and at the beginning with <code class="computeroutput"><span class="identifier">t0</span></code>, |
| similar as above. The approximate result 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>. This function |
| 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>. |
| </p> |
| <h5> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h2"></a> |
| <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 |
| calls at each step</a> |
| </h5> |
| <p> |
| 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. |
| Note that in the case of <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> or <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense |
| Output Stepper</a> this leads to non-equidistant observer calls as the |
| step size changes. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_adaptive</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> |
| with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. |
| Integration start at <code class="computeroutput"><span class="identifier">t0</span></code> and |
| <code class="computeroutput"><span class="identifier">x0</span></code> and ends at <span class="emphasis"><em>t<sub>1</sub></em></span>. |
| <code class="computeroutput"><span class="identifier">x0</span></code> is changed to the approximative |
| solution <span class="emphasis"><em>x(t<sub>1</sub>)</em></span> at the end of integration. If provided, |
| the <code class="computeroutput"><span class="identifier">observer</span></code> is called after |
| each step (and before the first step at <code class="computeroutput"><span class="identifier">t0</span></code>). |
| <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> returns |
| the number of steps performed during the integration. |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 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> |
| then <code class="computeroutput"><span class="identifier">dt</span></code> is the step size |
| used for integration and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> |
| behaves like <code class="computeroutput"><span class="identifier">integrate_const</span></code> |
| except that for the last step the step size is reduced to ensure we end |
| exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. If provided, |
| the observer is called at each step. |
| </li> |
| <li class="listitem"> |
| If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size. The actual step size is changed according to |
| error control of the stepper. For the last step, the step size will be |
| reduced to ensure we end exactly at <code class="computeroutput"><span class="identifier">t1</span></code>. |
| If provided, the observer is called after each time step (and before |
| the first step at <code class="computeroutput"><span class="identifier">t0</span></code>). |
| </li> |
| <li class="listitem"> |
| If stepper is a <a class="link" href="../concepts/dense_output_stepper.html" title="Dense Output Stepper">Dense |
| Output Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size and <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> |
| behaves just like for <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> above. No dense output is used. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h3"></a> |
| <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 |
| calls at given time points</a> |
| </h5> |
| <p> |
| 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. |
| The times for observer calls are provided as a sequence of time values. The |
| sequence is either defined via two iterators pointing to begin and end of |
| the sequence or in terms of a <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a> |
| object. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">times_start</span> <span class="special">,</span> |
| <span class="identifier">times_end</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate_times</span><span class="special">(</span> |
| <span class="identifier">stepper</span> <span class="special">,</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">time_range</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| Integrates the ODE given by <code class="computeroutput"><span class="identifier">system</span></code> |
| with subsequent steps from <code class="computeroutput"><span class="identifier">stepper</span></code>. |
| Integration starts at <code class="computeroutput"><span class="special">*</span><span class="identifier">times_start</span></code> |
| 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>. |
| <code class="computeroutput"><span class="identifier">x0</span></code> contains the approximate |
| solution at the end point of integration. This function requires an observer |
| 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> |
| until <code class="computeroutput"><span class="identifier">times_start</span> <span class="special">==</span> |
| <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> |
| <code class="computeroutput"><span class="identifier">time_range</span></code> the function behaves |
| 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> |
| <span class="special">)</span></code> and <code class="computeroutput"><span class="identifier">times_end</span> |
| <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">end</span><span class="special">(</span> |
| <span class="identifier">time_range</span> <span class="special">)</span></code>. |
| <code class="computeroutput"><span class="identifier">integrate_times</span></code> returns the |
| number of steps performed during the integration. |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 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> |
| <code class="computeroutput"><span class="identifier">dt</span></code> is the step size used |
| for integration. However, whenever a time point from the sequence is |
| approached the step size <code class="computeroutput"><span class="identifier">dt</span></code> |
| will be reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> exactly |
| at the time point. |
| </li> |
| <li class="listitem"> |
| If <code class="computeroutput"><span class="identifier">stepper</span></code> is a <a class="link" href="../concepts/controlled_stepper.html" title="Controlled Stepper">Controlled |
| Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size. The actual step size is adjusted during integration |
| according to error control. However, if a time point from the sequence |
| is approached the step size is reduced to obtain the state <span class="emphasis"><em>x(t)</em></span> |
| exactly at the time point. |
| </li> |
| <li class="listitem"> |
| 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 |
| Stepper</a> then <code class="computeroutput"><span class="identifier">dt</span></code> |
| is the initial step size. The actual step size is adjusted during integration |
| according to error control. Dense output is used to obtain the states |
| <span class="emphasis"><em>x(t)</em></span> at the time points from the sequence. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="boost_numeric_odeint.odeint_in_detail.integrate_functions.h4"></a> |
| <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 |
| integrate function</a> |
| </h5> |
| <p> |
| Additionally to the sophisticated integrate function above odeint also provides |
| a simple <code class="computeroutput"><span class="identifier">integrate</span></code> routine |
| which uses a dense output stepper based on <code class="computeroutput"><span class="identifier">runge_kutta_dopri5</span></code> |
| with standard error bounds <span class="emphasis"><em>10<sup>-6</sup></em></span> for the steps. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">integrate</span><span class="special">(</span> |
| <span class="identifier">system</span> <span class="special">,</span> |
| <span class="identifier">x0</span> <span class="special">,</span> |
| <span class="identifier">t0</span> <span class="special">,</span> |
| <span class="identifier">t1</span> <span class="special">,</span> |
| <span class="identifier">dt</span> <span class="special">,</span> |
| <span class="identifier">observer</span> <span class="special">)</span></code> |
| </p> |
| <p> |
| This function behaves exactly like <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> |
| above but no stepper has to be provided. It also returns the number of steps |
| performed during the integration. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2009-2012 Karsten |
| Ahnert and Mario Mulansky<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
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