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### Controlled Stepper

This concept specifies the interface a controlled stepper has to fulfill to be used within integrate functions.

##### Description

A controlled stepper following this Controlled Stepper concept provides the possibility to perform one step of the solution x(t) of an ODE with step-size dt to obtain x(t+dt) with a given step-size dt. Depending on an error estimate of the solution the step might be rejected and a smaller step-size is suggested.

##### Associated types

- state_type

  `Stepper::state_type`

  The type characterizing the state of the ODE, hence x.

- deriv_type

  `Stepper::deriv_type`

  The type characterizing the derivative of the ODE, hence d x/dt.

- time_type

  `Stepper::time_type`

  The type characterizing the dependent variable of the ODE, hence the time t.

- value_type

  `Stepper::value_type`

  The numerical data type which is used within the stepper, something like `float`, `double`, `complex&lt; double &gt;`.

- stepper_category

  `Stepper::stepper_category`

  A tag type characterizing the category of the stepper. This type must be convertible to `controlled_stepper_tag`.

##### Notation

`ControlledStepper`

  A type that is a model of Controlled 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 `ControlledStepper`

`x`

  Object of type `State`

`t`, `dt`

  Objects of type `Time`

`sys`

  An object defining the ODE, should be a model of System, Symplectic System, Simple Symplectic System or Implicit System.

##### Valid Expressions

Name | Expression | Type | Semantics
--- | --- | --- | ---

Do step

    stepper.try_step( sys , x , t , dt )

`controlled_step_result`

Tries one step of step size `dt`. If the step was successful, `success` is returned, the resulting state is written to `x`, the new time is stored in `t` and `dt` now contains a new (possibly larger) step-size for the next step. If the error was too big, `rejected` is returned and the results are neglected - `x` and `t` are unchanged and `dt` now contains a reduced step-size to be used for the next try.

##### Models

- `controlled_error_stepper< runge_kutta_cash_karp54 >`
- `controlled_error_stepper_fsal< runge_kutta_dopri5 >`
- `controlled_error_stepper< runge_kutta_fehlberg78 >`
- `rosenbrock4_controller`
- `bulirsch_stoer`