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

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.


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).


Initial time.

y0array_like, shape (n,)

Initial state.


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

first_stepfloat or None, optional

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

max_stepfloat, optional

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

rtol, atolfloat 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.

vectorizedbool, optional

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



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.


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


Number of equations.


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


Boundary time.


Integration direction: +1 or -1.


Current time.


Current state.


Previous time. None if no steps were made yet.


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


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


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


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



Compute a local interpolant over the last successful step.


Perform one integration step.