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), while atol controls absolute accuracy (number of correct decimal places). To achieve the desired rtol, set atol to be lower than the lowest value that can be expected fromrtol * abs(y)
so that rtol dominates the allowable error. If atol is larger thanrtol * abs(y)
the number of correct digits is not guaranteed. Conversely, to achieve the desired atol set rtol such thatrtol * abs(y)
is always lower than atol. 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.