DOP853(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 8.
This is a Python implementation of “DOP853” algorithm originally written in Fortran [R2069d09a2148-1], [R2069d09a2148-2]. Note that this is not a literate translation, but the algorithmic core and coefficients are the same.
Can be applied in the complex domain.
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).
- y0array_like, shape (n,)
Boundary time - the integration won’t continue beyond it. It also determines the direction of the integration.
- first_stepfloat or None, optional
Initial step size. Default is
Nonewhich means that the algorithm should choose.
- max_stepfloat, optional
Maximum allowed step size. Default is np.inf, i.e. the step size is not bounded and determined solely by the solver.
- rtol, atolfloat 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.
- vectorizedbool, optional
Whether fun is implemented in a vectorized fashion. Default is False.
E. Hairer, S. P. Norsett G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems”, Sec. II.
Number of equations.
Current status of the solver: ‘running’, ‘finished’ or ‘failed’.
Integration direction: +1 or -1.
Previous time. None if no steps were made yet.
Size of the last successful step. None if no steps were made yet.
Number evaluations of the system’s right-hand side.
Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
Number of LU decompositions. Is always 0 for this solver.
Compute a local interpolant over the last successful step.
Perform one integration step.