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## Class template runge_kutta_fehlberg78

boost::numeric::odeint::runge_kutta_fehlberg78 — The Runge-Kutta Fehlberg 78 method.

## Synopsis

    // In header: <boost/numeric/odeint/stepper/runge_kutta_fehlberg78.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 runge_kutta_fehlberg78 : public boost::numeric::odeint::explicit_error_generic_rk< StageCount, Order, StepperOrder, ErrorOrder, State, Value, Deriv, Time, Algebra, Operations, Resizer >
    {
    public:
      // types
      typedef stepper_base_type::state_type      state_type;
      typedef stepper_base_type::value_type      value_type;
      typedef stepper_base_type::deriv_type      deriv_type;
      typedef stepper_base_type::time_type       time_type;
      typedef stepper_base_type::algebra_type    algebra_type;
      typedef stepper_base_type::operations_type operations_type;
      typedef stepper_base_type::resizer_type    resizer_type;

      // construct/copy/destruct
      runge_kutta_fehlberg78(const algebra_type & = algebra_type());

      // public member functions
      template<typename System, typename StateIn, typename DerivIn,
               typename StateOut, typename Err>
        void do_step_impl(System, const StateIn &, const DerivIn &, time_type,
                          StateOut &, time_type, Err &);
      template<typename System, typename StateIn, typename DerivIn,
               typename StateOut>
        void do_step_impl(System, const StateIn &, const DerivIn &, time_type,
                          StateOut &, time_type);
      template<typename StateIn> void adjust_size(const StateIn &);
    };

## Description

The Runge-Kutta Fehlberg 78 method is a standard method for high-precision applications. The method is explicit and fulfills the Error Stepper concept. Step size control is provided but continuous output is not available for this method.

This class derives from explicit_error_stepper_base and inherits its interface via CRTP (current recurring template pattern). Furthermore, it derivs from explicit_error_generic_rk which is a generic Runge-Kutta algorithm with error estimation. For more details see explicit_error_stepper_base and explicit_error_generic_rk.

### 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.

### `runge_kutta_fehlberg78` public construct/copy/destruct

1.     runge_kutta_fehlberg78(const algebra_type & algebra = algebra_type());

   Constructs the runge_kutta_cash_fehlberg78 class. This constructor can be used as a default constructor if the algebra has a default constructor.

   Parameters: |
   `algebra` | A copy of algebra is made and stored inside explicit_stepper_base.

### `runge_kutta_fehlberg78` public member functions

1.     template<typename System, typename StateIn, typename DerivIn,
                typename StateOut, typename Err>
         void do_step_impl(System system, const StateIn & in, const DerivIn & dxdt,
                           time_type t, StateOut & out, time_type dt, Err & xerr);

   This method performs one step. The derivative `dxdt` of `in` at the time `t` is passed to the method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Futhermore, an estimation of the error is stored in `xerr`. `do_step_impl` is used by explicit_error_stepper_base.

   Parameters: |
   `dt` | The step size.
   `dxdt` | The derivative of x at t.
   `in` | The state of the ODE which should be solved. in is not modified in this method
   `out` | The result of the step is written in out.
   `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, at which the step should be performed.
   `xerr` | The result of the error estimation is written in xerr.

2.     template<typename System, typename StateIn, typename DerivIn,
                typename StateOut>
         void do_step_impl(System system, const StateIn & in, const DerivIn & dxdt,
                           time_type t, StateOut & out, time_type dt);

   This method performs one step. The derivative `dxdt` of `in` at the time `t` is passed to the method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Access to this step functionality is provided by explicit_stepper_base and `do_step_impl` should not be called directly.

   Parameters: |
   `dt` | The step size.
   `dxdt` | The derivative of x at t.
   `in` | The state of the ODE which should be solved. in is not modified in this method
   `out` | The result of the step is written in out.
   `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, at which the step should be performed.

3.     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.