scipy.integrate.RK45¶
- class scipy.integrate.RK45(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, vectorized=False, first_step=None, **extraneous)[source]¶
Explicit Runge-Kutta method of order 5(4).
This uses the Dormand-Prince pair of formulas [1]. 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 [2].
Can be applied in the complex domain.
- Parameters
- funcallable
Right-hand 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 1e-3 for rtol and 1e-6 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 Runge-Kutta formulae”, Journal of Computational and Applied Mathematics, Vol. 6, No. 1, pp. 19-26, 1980.
- 2
L. W. Shampine, “Some Practical Runge-Kutta Formulas”, Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 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 right-hand 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
Compute a local interpolant over the last successful step.
step
()Perform one integration step.