scipy.integrate.RK45¶

class
scipy.integrate.
RK45
(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e06, vectorized=False, **extraneous)[source]¶ Explicit RungeKutta method of order 5(4).
The DormandPrince 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.
Parameters: fun : callable
Righthand side of the system. The calling signature is
fun(t, y)
. Heret
is a scalar and there are two options for ndarrayy
. It can either have shape (n,), thenfun
must return array_like with shape (n,). Or alternatively it can have shape (n, k), thenfun
must return array_like with shape (n, k), i.e. each column corresponds to a single column iny
. 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 1e3 for rtol and 1e6 for atol.vectorized : bool, optional
Whether fun is implemented in a vectorized fashion. Default is False.
References
[R59] (1, 2) J. R. Dormand, P. J. Prince, “A family of embedded RungeKutta formulae”, Journal of Computational and Applied Mathematics, Vol. 6, No. 1, pp. 1926, 1980. [R60] (1, 2) L. W. Shampine, “Some Practical RungeKutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135150, 1986. 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 of the system’s rhs evaluations. njev (int) Number of the Jacobian evaluations. nlu (int) Number of LU decompositions. Methods
dense_output
()Compute a local interpolant over the last successful step. step
()Perform one integration step.