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Class template bulirsch_stoer

boost::numeric::odeint::bulirsch_stoer — The Bulirsch-Stoer algorithm.

Synopsis

// In header: <boost/numeric/odeint/stepper/bulirsch_stoer.hpp>

template<typename State, typename Value = double, typename Deriv = State, 
         typename Time = Value, 
         typename Algebra = typename algebra_dispatcher< State >::algebra_type, 
         typename Operations = typename operations_dispatcher< State >::operations_type, 
         typename Resizer = initially_resizer> 
class bulirsch_stoer {
public:
  // types
  typedef State      state_type;     
  typedef Value      value_type;     
  typedef Deriv      deriv_type;     
  typedef Time       time_type;      
  typedef Algebra    algebra_type;   
  typedef Operations operations_type;
  typedef Resizer    resizer_type;   

  // construct/copy/destruct
  bulirsch_stoer(value_type = 1E-6, value_type = 1E-6, value_type = 1.0, 
                 value_type = 1.0, time_type = static_cast< time_type >(0));

  // public member functions
  template<typename System, typename StateInOut> 
    controlled_step_result 
    try_step(System, StateInOut &, time_type &, time_type &);
  template<typename System, typename StateInOut> 
    controlled_step_result 
    try_step(System, const StateInOut &, time_type &, time_type &);
  template<typename System, typename StateInOut, typename DerivIn> 
    controlled_step_result 
    try_step(System, StateInOut &, const DerivIn &, time_type &, time_type &);
  template<typename System, typename StateIn, typename StateOut> 
    boost::disable_if< boost::is_same< StateIn, time_type >, controlled_step_result >::type 
    try_step(System, const StateIn &, time_type &, StateOut &, time_type &);
  template<typename System, typename StateIn, typename DerivIn, 
           typename StateOut> 
    controlled_step_result 
    try_step(System, const StateIn &, const DerivIn &, time_type &, 
             StateOut &, time_type &);
  void reset();
  template<typename StateIn> void adjust_size(const StateIn &);

  // private member functions
  template<typename StateIn> bool resize_m_dxdt(const StateIn &);
  template<typename StateIn> bool resize_m_xnew(const StateIn &);
  template<typename StateIn> bool resize_impl(const StateIn &);
  template<typename System, typename StateInOut> 
    controlled_step_result 
    try_step_v1(System, StateInOut &, time_type &, time_type &);
  template<typename StateInOut> 
    void extrapolate(size_t, state_table_type &, const value_matrix &, 
                     StateInOut &);
  time_type calc_h_opt(time_type, value_type, size_t) const;
  controlled_step_result 
  set_k_opt(size_t, const inv_time_vector &, const time_vector &, time_type &);
  bool in_convergence_window(size_t) const;
  bool should_reject(value_type, size_t) const;

  // public data members
  static const size_t m_k_max;
};

Description

The Bulirsch-Stoer is a controlled stepper that adjusts both step size and order of the method. The algorithm uses the modified midpoint and a polynomial extrapolation compute the solution.

Template Parameters

  1. typename State

    The state type.

  2. typename Value = double

    The value type.

  3. typename Deriv = State

    The type representing the time derivative of the state.

  4. typename Time = Value

    The time representing the independent variable - the time.

  5. typename Algebra = typename algebra_dispatcher< State >::algebra_type

    The algebra type.

  6. typename Operations = typename operations_dispatcher< State >::operations_type

    The operations type.

  7. typename Resizer = initially_resizer

    The resizer policy type.

bulirsch_stoer public construct/copy/destruct

  1. bulirsch_stoer(value_type eps_abs = 1E-6, value_type eps_rel = 1E-6, 
                   value_type factor_x = 1.0, value_type factor_dxdt = 1.0, 
                   time_type max_dt = static_cast< time_type >(0));
    Constructs the bulirsch_stoer class, including initialization of the error bounds.

    Parameters:

    eps_abs

    Absolute tolerance level.

    eps_rel

    Relative tolerance level.

    factor_dxdt

    Factor for the weight of the derivative.

    factor_x

    Factor for the weight of the state.

bulirsch_stoer public member functions

  1. template<typename System, typename StateInOut> 
      controlled_step_result 
      try_step(System system, StateInOut & x, time_type & t, time_type & dt);
    Tries to perform one step.

    This method tries to do one step with step size dt. If the error estimate is to large, the step is rejected and the method returns fail and the step size dt is reduced. If the error estimate is acceptably small, the step is performed, success is returned and dt might be increased to make the steps as large as possible. This method also updates t if a step is performed. Also, the internal order of the stepper is adjusted if required.

    Parameters:

    dt

    The step size. Updated.

    system

    The system function to solve, hence the r.h.s. of the ODE. It must fulfill the Simple System concept.

    t

    The value of the time. Updated if the step is successful.

    x

    The state of the ODE which should be solved. Overwritten if the step is successful.

    Returns:

    success if the step was accepted, fail otherwise.

  2. template<typename System, typename StateInOut> 
      controlled_step_result 
      try_step(System system, const StateInOut & x, time_type & t, time_type & dt);
    Second version to solve the forwarding problem, can be used with Boost.Range as StateInOut.
  3. template<typename System, typename StateInOut, typename DerivIn> 
      controlled_step_result 
      try_step(System system, StateInOut & x, const DerivIn & dxdt, time_type & t, 
               time_type & dt);
    Tries to perform one step.

    This method tries to do one step with step size dt. If the error estimate is to large, the step is rejected and the method returns fail and the step size dt is reduced. If the error estimate is acceptably small, the step is performed, success is returned and dt might be increased to make the steps as large as possible. This method also updates t if a step is performed. Also, the internal order of the stepper is adjusted if required.

    Parameters:

    dt

    The step size. Updated.

    dxdt

    The derivative of state.

    system

    The system function to solve, hence the r.h.s. of the ODE. It must fulfill the Simple System concept.

    t

    The value of the time. Updated if the step is successful.

    x

    The state of the ODE which should be solved. Overwritten if the step is successful.

    Returns:

    success if the step was accepted, fail otherwise.

  4. template<typename System, typename StateIn, typename StateOut> 
      boost::disable_if< boost::is_same< StateIn, time_type >, controlled_step_result >::type 
      try_step(System system, const StateIn & in, time_type & t, StateOut & out, 
               time_type & dt);
    Tries to perform one step.
    [Note] Note

    This method is disabled if state_type=time_type to avoid ambiguity.

    This method tries to do one step with step size dt. If the error estimate is to large, the step is rejected and the method returns fail and the step size dt is reduced. If the error estimate is acceptably small, the step is performed, success is returned and dt might be increased to make the steps as large as possible. This method also updates t if a step is performed. Also, the internal order of the stepper is adjusted if required.

    Parameters:

    dt

    The step size. Updated.

    in

    The state of the ODE which should be solved.

    out

    Used to store the result of the step.

    system

    The system function to solve, hence the r.h.s. of the ODE. It must fulfill the Simple System concept.

    t

    The value of the time. Updated if the step is successful.

    Returns:

    success if the step was accepted, fail otherwise.

  5. template<typename System, typename StateIn, typename DerivIn, 
             typename StateOut> 
      controlled_step_result 
      try_step(System system, const StateIn & in, const DerivIn & dxdt, 
               time_type & t, StateOut & out, time_type & dt);
    Tries to perform one step.

    This method tries to do one step with step size dt. If the error estimate is to large, the step is rejected and the method returns fail and the step size dt is reduced. If the error estimate is acceptably small, the step is performed, success is returned and dt might be increased to make the steps as large as possible. This method also updates t if a step is performed. Also, the internal order of the stepper is adjusted if required.

    Parameters:

    dt

    The step size. Updated.

    dxdt

    The derivative of state.

    in

    The state of the ODE which should be solved.

    out

    Used to store the result of the step.

    system

    The system function to solve, hence the r.h.s. of the ODE. It must fulfill the Simple System concept.

    t

    The value of the time. Updated if the step is successful.

    Returns:

    success if the step was accepted, fail otherwise.

  6. void reset();
    Resets the internal state of the stepper.
  7. template<typename StateIn> void adjust_size(const StateIn & x);
    Adjust the size of all temporaries in the stepper manually.

    Parameters:

    x

    A state from which the size of the temporaries to be resized is deduced.

bulirsch_stoer private member functions

  1. template<typename StateIn> bool resize_m_dxdt(const StateIn & x);
  2. template<typename StateIn> bool resize_m_xnew(const StateIn & x);
  3. template<typename StateIn> bool resize_impl(const StateIn & x);
  4. template<typename System, typename StateInOut> 
      controlled_step_result 
      try_step_v1(System system, StateInOut & x, time_type & t, time_type & dt);
  5. template<typename StateInOut> 
      void extrapolate(size_t k, state_table_type & table, 
                       const value_matrix & coeff, StateInOut & xest);
  6. time_type calc_h_opt(time_type h, value_type error, size_t k) const;
  7. controlled_step_result 
    set_k_opt(size_t k, const inv_time_vector & work, const time_vector & h_opt, 
              time_type & dt);
  8. bool in_convergence_window(size_t k) const;
  9. bool should_reject(value_type error, size_t k) const;

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