RK45(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, vectorized=False, first_step=None, **extraneous)¶
Explicit Runge-Kutta method of order 5(4).
This uses the Dormand-Prince pair of formulas . 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 .
Can be applied in the complex domain.
- fun : callable
Right-hand side of the system. The calling signature is
fun(t, y). Here
tis a scalar, and there are two options for the ndarray
y: It can either have shape (n,); then
funmust return array_like with shape (n,). Alternatively it can have shape (n, k); then
funmust 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).
- t0 : float
- y0 : array_like, shape (n,)
- 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
Nonewhich 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 1e-3 for rtol and 1e-6 for atol.
- vectorized : bool, optional
Whether fun is implemented in a vectorized fashion. Default is False.
 (1, 2) 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.  (1, 2) L. W. Shampine, “Some Practical Runge-Kutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 1986. Attributes:
- n : int
Number of equations.
- status : string
Current status of the solver: ‘running’, ‘finished’ or ‘failed’.
- t_bound : float
- direction : float
Integration direction: +1 or -1.
- t : float
- y : ndarray
- 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.
Compute a local interpolant over the last successful step.
Perform one integration step.