class scipy.integrate.RK23(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, vectorized=False, **extraneous)[source]

Explicit Runge-Kutta method of order 3(2).

This uses the Bogacki-Shampine pair of formulas [1]. The error is controlled assuming accuracy of the second-order method, but steps are taken using the third-order accurate formula (local extrapolation is done). A cubic Hermite polynomial is used for the dense output.

Can be applied in the 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).

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.

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). But if a component of y is approximately below atol, 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.


[1](1, 2) P. Bogacki, L.F. Shampine, “A 3(2) Pair of Runge-Kutta Formulas”, Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
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.


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