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
 funcallable
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). t0float
Initial time.
 y0array_like, shape (n,)
Initial state.
 t_boundfloat
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
None
which 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 1e3 for rtol and 1e6 for atol. vectorizedbool, optional
Whether fun is implemented in a vectorized fashion. Default is False.
References
 1
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
L. W. Shampine, “Some Practical RungeKutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135150, 1986.
 Attributes
 nint
Number of equations.
 statusstring
Current status of the solver: ‘running’, ‘finished’ or ‘failed’.
 t_boundfloat
Boundary time.
 directionfloat
Integration direction: +1 or 1.
 tfloat
Current time.
 yndarray
Current state.
 t_oldfloat
Previous time. None if no steps were made yet.
 step_sizefloat
Size of the last successful step. None if no steps were made yet.
 nfevint
Number evaluations of the system’s righthand side.
 njevint
Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
 nluint
Number of LU decompositions. Is always 0 for this solver.
Methods
dense_output
(self)Compute a local interpolant over the last successful step.
step
(self)Perform one integration step.