class scipy.integrate.RK45(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, vectorized=False, **extraneous)[source]

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

The Dormand-Prince pair of formulas is used [R59]. The error is controlled assuming 4th order accuracy, but steps are taken using a 5th oder accurate formula (local extrapolation is done). A quartic interpolation polynomial is used for the dense output [R60].

Can be applied in a complex domain.


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 ndarray y. It can either have shape (n,), then fun must return array_like with shape (n,). Or alternatively it can have shape (n, k), then fun must return 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). The vectorized implementation allows faster approximation of the Jacobian by finite differences.

t0 : float

Initial value of the independent variable.

y0 : array_like, shape (n,)

Initial values of the dependent variable.

t_bound : float

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

max_step : float, optional

Maximum allowed step size. Default is np.inf, i.e. the step 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). But if a component of y is approximately below atol then the error only needs to fall within the same atol threshold, and the number of correct digits is not guaranteed. 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.


[R59](1, 2) 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.
[R60](1, 2) L. W. Shampine, “Some Practical Runge-Kutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.


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 of the system’s rhs evaluations.
njev (int) Number of the Jacobian evaluations.
nlu (int) Number of LU decompositions.


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