Implicit Runge-Kutta method of Radau IIA family of order 5.
The implementation follows 1_. The error is controlled with a third-order accurate embedded formula. A cubic polynomial which satisfies the collocation conditions is used for the dense output.
Parameters ---------- fun : callable Right-hand side of the system. The calling signature is ``fun(t, y)``. Here ``t`` is a scalar, and there are two options for the ndarray ``y``: It can either have shape (n,); then ``fun`` must return array_like with shape (n,). Alternatively it can have shape (n, k); then ``fun`` must 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). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for this solver). t0 : float Initial time. y0 : array_like, shape (n,) Initial state. 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 ``None`` which 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`. jac : None, array_like, sparse_matrix, callable, optional Jacobian matrix of the right-hand side of the system with respect to y, required by this method. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``. There are three ways to define the Jacobian:
* If array_like or sparse_matrix, the Jacobian is assumed to be constant. * If callable, the Jacobian is assumed to depend on both t and y; it will be called as ``jac(t, y)`` as necessary. For the 'Radau' and 'BDF' methods, the return value might be a sparse matrix. * If None (default), the Jacobian will be approximated by finite differences.
It is generally recommended to provide the Jacobian rather than relying on a finite-difference approximation. jac_sparsity : None, array_like, sparse matrix, optional Defines a sparsity structure of the Jacobian matrix for a finite-difference approximation. Its shape must be (n, n). This argument is ignored if `jac` is not `None`. If the Jacobian has only few non-zero elements in *each* row, providing the sparsity structure will greatly speed up the computations 2_. A zero entry means that a corresponding element in the Jacobian is always zero. If None (default), the Jacobian is assumed to be dense. vectorized : bool, optional Whether `fun` is implemented in a vectorized fashion. Default is False.
Attributes ---------- n : int Number of equations. status : string Current status of the solver: 'running', 'finished' or 'failed'. t_bound : float Boundary time. direction : float Integration direction: +1 or -1. t : float Current time. y : ndarray Current state. 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 of evaluations of the right-hand side. njev : int Number of evaluations of the Jacobian. nlu : int Number of LU decompositions.
References ---------- .. 1 E. Hairer, G. Wanner, 'Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems', Sec. IV.8. .. 2 A. Curtis, M. J. D. Powell, and J. Reid, 'On the estimation of sparse Jacobian matrices', Journal of the Institute of Mathematics and its Applications, 13, pp. 117-120, 1974.