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scipy.integrate.

# RK45

class scipy.integrate.RK45(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, vectorized=False, first_step=None, **extraneous)

  Explicit Runge-Kutta method of order 5(4).

  This uses the Dormand-Prince pair of formulas [1]. The error is controlled assuming accuracy of the fourth-order method accuracy, but steps are taken using the fifth-order accurate formula (local extrapolation is done). A quartic interpolation polynomial is used for the dense output [2].

  Can be applied in the complex domain.

  Parameters:
    fun callable

      Right-hand side of the system. The calling signature is `fun(t, y)`. Here `t` is a scalar, and there are two options for the ndarray `y`: It can either have shape (n,); then `fun` must return array_like with shape (n,). Alternatively it can have shape (n, k); then `fun` must return an array_like with shape (n, k), i.e., each column corresponds to a single column in `y`. The choice between the two options is determined by vectorized argument (see below).

    t0 float

      Initial time.

    y0 array_like, shape (n,)

      Initial state.

    t_bound float

      Boundary time - the integration won’t continue beyond it. It also determines the direction of the integration.

    first_step float or None, optional

      Initial step size. Default is `None` which means that the algorithm should choose.

    max_step float, optional

      Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver.

    rtol, atol float and array_like, optional

      Relative and absolute tolerances. The solver keeps the local error estimates less than `atol + rtol * abs(y)`. Here rtol controls a relative accuracy (number of correct digits), while atol controls absolute accuracy (number of correct decimal places). To achieve the desired rtol, set atol to be smaller than the smallest value that can be expected from `rtol * abs(y)` so that rtol dominates the allowable error. If atol is larger than `rtol * abs(y)` the number of correct digits is not guaranteed. Conversely, to achieve the desired atol set rtol such that `rtol * abs(y)` is always smaller than atol. If components of y have different scales, it might be beneficial to set different atol values for different components by passing array_like with shape (n,) for atol. Default values are 1e-3 for rtol and 1e-6 for atol.

    vectorized bool, optional

      Whether fun is implemented in a vectorized fashion. Default is False.

  Attributes:
    n int

      Number of equations.

    status string

      Current status of the solver: ‘running’, ‘finished’ or ‘failed’.

    t_bound float

      Boundary time.

    direction float

      Integration direction: +1 or -1.

    t float

      Current time.

    y ndarray

      Current state.

    t_old float

      Previous time. None if no steps were made yet.

    step_size float

      Size of the last successful step. None if no steps were made yet.

    nfev int

      Number evaluations of the system’s right-hand side.

    njev int

      Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.

    nlu int

      Number of LU decompositions. Is always 0 for this solver.

  Methods

  `dense_output`() | Compute a local interpolant over the last successful step.
  `step`() | Perform one integration step.

  References

  [1]

  J. R. Dormand, P. J. Prince, “A family of embedded Runge-Kutta formulae”, Journal of Computational and Applied Mathematics, Vol. 6, No. 1, pp. 19-26, 1980.

  [2]

  L. W. Shampine, “Some Practical Runge-Kutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.