scipy.integrate.RK23¶

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.

Parameters:

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 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.

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.

References

[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.

Attributes:

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.

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.

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