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## Class template euler

boost::numeric::odeint::euler — An implementation of the Euler method.

## Synopsis

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

    template<typename State, typename Value = double, typename Deriv = State,
             typename Time = Value, typename Algebra = range_algebra,
             typename Operations = default_operations,
             typename Resizer = initially_resizer>
    class euler : public boost::numeric::odeint::explicit_stepper_base< Stepper, Order, State, Value, Deriv, Time, Algebra, Operations, Resizer >
    {
    public:
      // types
      typedef explicit_stepper_base< euler< ... >,... > stepper_base_type;
      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
      euler(const algebra_type & = algebra_type());

      // public member functions
      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 StateOut, typename StateIn1, typename StateIn2>
        void calc_state(StateOut &, time_type, const StateIn1 &, time_type,
                        const StateIn2 &, time_type) const;
      template<typename StateType> void adjust_size(const StateType &);
      template<typename System, typename StateInOut>
        void do_step(System, StateInOut &, time_type, time_type);
      template<typename System, typename StateInOut>
        void do_step(System, const StateInOut &, time_type, time_type);
      template<typename System, typename StateInOut, typename DerivIn>
        boost::disable_if< boost::is_same< DerivIn, time_type >, void >::type
        do_step(System, StateInOut &, const DerivIn &, time_type, time_type);
      template<typename System, typename StateIn, typename StateOut>
        void do_step(System, const StateIn &, time_type, StateOut &, time_type);
      template<typename System, typename StateIn, typename DerivIn,
               typename StateOut>
        void do_step(System, const StateIn &, const DerivIn &, time_type,
                     StateOut &, time_type);
      template<typename StateIn> void adjust_size(const StateIn &);
      const algebra_type & algebra() const;
    };

## Description

The Euler method is a very simply solver for ordinary differential equations. This method should not be used for real applications. It is only useful for demonstration purposes. Step size control is not provided but trivial continuous output is available.

This class derives from explicit_stepper_base and inherits its interface via CRTP (current recurring template pattern), see explicit_stepper_base

### 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 = range_algebra

   The algebra type.

6.     typename Operations = default_operations

   The operations type.

7.     typename Resizer = initially_resizer

   The resizer policy type.

### `euler` public construct/copy/destruct

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

   Constructs the euler class. This constructor can be used as a default constructor of the algebra has a default constructor.

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

### `euler` public member functions

1.     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.

2.     template<typename StateOut, typename StateIn1, typename StateIn2>
         void calc_state(StateOut & x, time_type t, const StateIn1 & old_state,
                         time_type t_old, const StateIn2 & current_state,
                         time_type t_new) const;

   This method is used for continuous output and it calculates the state `x` at a time `t` from the knowledge of two states `old_state` and `current_state` at time points `t_old` and `t_new`.

3.     template<typename StateType> void adjust_size(const StateType & 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.

4.     template<typename System, typename StateInOut>
         void do_step(System system, StateInOut & x, time_type t, time_type dt);

   This method performs one step. It transforms the result in-place.

   Parameters: |
   `dt` | The step size.
   `system` | The system function to solve, hence the r.h.s. of the ordinary differential equation. It must fulfill the Simple System concept.
   `t` | The value of the time, at which the step should be performed.
   `x` | The state of the ODE which should be solved. After calling do_step the result is updated in x.

5.     template<typename System, typename StateInOut>
         void do_step(System system, const StateInOut & x, time_type t, time_type dt);

   Second version to solve the forwarding problem, can be called with Boost.Range as StateInOut.

6.     template<typename System, typename StateInOut, typename DerivIn>
         boost::disable_if< boost::is_same< DerivIn, time_type >, void >::type
         do_step(System system, StateInOut & x, const DerivIn & dxdt, time_type t,
                 time_type dt);

   The method performs one step. Additionally to the other method the derivative of x is also passed to this method. It is supposed to be used in the following way:

       * sys( x , dxdt , t );
       * stepper.do_step( sys , x , dxdt , t , dt );
       *

   The result is updated in place in x. This method is disabled if Time and Deriv are of the same type. In this case the method could not be distinguished from other `do_step` versions.

   | Note
   This method does not solve the forwarding problem.

   Parameters: |
   `dt` | The step size.
   `dxdt` | The derivative of x at t.
   `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.
   `x` | The state of the ODE which should be solved. After calling do_step the result is updated in x.

7.     template<typename System, typename StateIn, typename StateOut>
         void do_step(System system, const StateIn & in, time_type t, StateOut & out,
                      time_type dt);

   The method performs one step. The state of the ODE is updated out-of-place.

   | Note
   This method does not solve the forwarding problem.

   Parameters: |
   `dt` | The step size.
   `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.

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

   The method performs one step. The state of the ODE is updated out-of-place. Furthermore, the derivative of x at t is passed to the stepper. It is supposed to be used in the following way:

       * sys( in , dxdt , t );
       * stepper.do_step( sys , in , dxdt , t , out , dt );
       *

   | Note
   This method does not solve the forwarding problem.

   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.

9.     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.

10.     const algebra_type & algebra() const;

    Returns: | A reference to the algebra which is held by this class. A const reference to the algebra which is held by this class.