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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, 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
- exception if too many steps are taken between observer calls (i.e. too small
- step size). 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>
- <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>
- <span class="identifier">max_step_checker</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>. If provided,
- the <code class="computeroutput"><span class="identifier">max_step_checker</span></code> counts
- the number of steps between observer calls and throws a <code class="computeroutput"><span class="identifier">no_progress_error</span></code>
- this exceeds some limit (default: 500). <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>
- <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> <span class="identifier">max_step_checker</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. Again, providing a <code class="computeroutput"><span class="identifier">max_step_checker</span></code>
- will throw a <code class="computeroutput"><span class="identifier">no_progress_error</span></code>
- if too many steps are performed between observer calls. 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="note"><table border="0" summary="Note">
- <tr>
- <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
- <th align="left">Note</th>
- </tr>
- <tr><td align="left" valign="top"><p>
- <code class="computeroutput"><span class="identifier">integrate_adaptive</span></code> by design
- performs an observer call after each time step. Hence there is no need
- 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
- exactly one step is ever performed between observer calls.
- </p></td></tr>
- </table></div>
- <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>.
- 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,
- e.g.:
- </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>
- <span class="identifier">max_step_checker</span><span class="special">)</span></code>
- </p>
- <p>
- As above, this will throw a <code class="computeroutput"><span class="identifier">no_progress_error</span></code>
- if too many steps are performed between observer calls.
- </p>
- <p>
- <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 © 2009-2015 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>
- <hr>
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