[/============================================================================
  Boost.odeint

  Copyright 2011-2012 Karsten Ahnert
  Copyright 2011 Mario Mulansky
  Copyright 2012 Sylwester Arabas

  Use, modification and distribution is subject to the Boost Software License,
  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt)
=============================================================================/]


[section Stepper]

This concepts specifies the interface a simple stepper has to fulfill to be used within the __integrate_functions.

[heading Description]

The basic stepper concept.
A basic stepper following this Stepper concept is able to perform a single step of the solution /x(t)/ of an ODE to obtain /x(t+dt)/ using a given step size /dt/.
Basic steppers can be Runge-Kutta steppers, symplectic steppers as well as implicit steppers.
Depending on the actual stepper, the ODE is defined as __system, __symplectic_system, __simple_symplectic_system or __implicit_system.
Note that all error steppers are also basic steppers.

[heading Refinement of]

* DefaultConstructable
* CopyConstructable


[heading Associated types]

*  '''<para>'''[*state_type]'''</para>'''
'''<para>'''`Stepper::state_type`'''</para>'''
'''<para>'''The type characterizing the state of the ODE, hence ['x].'''</para>'''

*  '''<para>'''[*deriv_type]'''</para>'''
'''<para>'''`Stepper::deriv_type`'''</para>'''
'''<para>'''The type characterizing the derivative of the ODE, hence ['d x/dt].'''</para>'''

*  '''<para>'''[*time_type]'''</para>'''
'''<para>'''`Stepper::time_type`'''</para>'''
'''<para>'''The type characterizing the dependent variable of the ODE, hence the time ['t].'''</para>'''

*  '''<para>'''[*value_type]'''</para>'''
'''<para>'''`Stepper::value_type`'''</para>'''
'''<para>'''The numerical data type which is used within the stepper, something like `float`, `double`, `complex&lt; double &gt;`.'''</para>'''

*  '''<para>'''[*order_type]'''</para>'''
'''<para>'''`Stepper::order_type`'''</para>'''
'''<para>'''The type characterizing the order of the ODE, typically `unsigned short`.'''</para>'''

*  '''<para>'''[*stepper_category]'''</para>'''
'''<para>'''`Stepper::stepper_category`'''</para>'''
'''<para>'''A tag type characterizing the category of the stepper. This type must be convertible to `stepper_tag`.'''</para>'''


[heading Notation]

[variablelist 
  [[`Stepper`] [A type that is a model of Stepper]]
  [[`State`] [A type representing the state /x/ of the ODE]]
  [[`Time`] [A type representing the time /t/ of the ODE]]
  [[`stepper`] [An object of type `Stepper`]]
  [[`x`] [Object of type `State`]]
  [[`t`, `dt`] [Objects of type `Time`]]
  [[`sys`] [An object defining the ODE. Depending on the Stepper this might be a model of __system, __symplectic_system, __simple_symplectic_system or __implicit_system ]]
]

[heading Valid Expressions]

[table
  [[Name] [Expression] [Type] [Semantics]]
  [[Get the order] [`stepper.order()`] [`order_type`] [Returns the order of the stepper.]]
  [[Do step] [`stepper.do_step( sys , x , t , dt )`] [`void`] [Performs one step of step size `dt`.  The newly obtained state is written in place in `x`.] ]

  [/ [Do step with reference] [`stepper.do_step( boost::ref(sys) , x , t , dt )`] [`void`] [Performs one step of step size `dt`.  The newly obtained state is written in place in `x`.] ]

  [/ [Do step out-of-place] [`stepper.do_step( sys , in , t , out , dt )`] [`void`] [Performs one step. The newly obtained state is written to `out`] ]
]

[heading Models]

* `runge_kutta4`
* `euler`
* `runge_kutta_cash_karp54`
* `runge_kutta_dopri5`
* `runge_kutta_fehlberg78`
* `modified_midpoint`
* `rosenbrock4`

[endsect]
