scipy.integrate.RK45¶

class
scipy.integrate.
RK45
(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e06, vectorized=False, first_step=None, **extraneous)[source]¶ Explicit RungeKutta method of order 5(4).
This uses the DormandPrince pair of formulas [1]. The error is controlled assuming accuracy of the fourthorder method accuracy, but steps are taken using the fifthorder accurate formula (local extrapolation is done). A quartic interpolation polynomial is used for the dense output [2].
Can be applied in the 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 the ndarrayy
: It can either have shape (n,); thenfun
must return array_like with shape (n,). Alternatively it can have shape (n, k); thenfun
must return an 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). 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.
 first_step : float or None, optional
Initial step size. Default is
None
which means that the algorithm should choose. 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 1e3 for rtol and 1e6 for atol. vectorized : bool, optional
Whether fun is implemented in a vectorized fashion. Default is False.
References
[1] (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. [2] (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 evaluations of the system’s righthand 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.