Base class for ODE solvers.
In order to implement a new solver you need to follow the guidelines:
1. A constructor must accept parameters presented in the base class (listed below) along with any other parameters specific to a solver. 2. A constructor must accept arbitrary extraneous arguments ``**extraneous``, but warn that these arguments are irrelevant using `common.warn_extraneous` function. Do not pass these arguments to the base class. 3. A solver must implement a private method `_step_impl(self)` which propagates a solver one step further. It must return tuple ``(success, message)``, where ``success`` is a boolean indicating whether a step was successful, and ``message`` is a string containing description of a failure if a step failed or None otherwise. 4. A solver must implement a private method `_dense_output_impl(self)` which returns a `DenseOutput` object covering the last successful step. 5. A solver must have attributes listed below in Attributes section. Note that ``t_old`` and ``step_size`` are updated automatically. 6. Use `fun(self, t, y)` method for the system rhs evaluation, this way the number of function evaluations (`nfev`) will be tracked automatically. 7. For convenience a base class provides `fun_single(self, t, y)` and `fun_vectorized(self, t, y)` for evaluating the rhs in non-vectorized and vectorized fashions respectively (regardless of how `fun` from the constructor is implemented). These calls don't increment `nfev`. 8. If a solver uses a Jacobian matrix and LU decompositions, it should track the number of Jacobian evaluations (`njev`) and the number of LU decompositions (`nlu`). 9. By convention the function evaluations used to compute a finite difference approximation of the Jacobian should not be counted in `nfev`, thus use `fun_single(self, t, y)` or `fun_vectorized(self, t, y)` when computing a finite difference approximation of the Jacobian.
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 ndarray ``y``. It can either have shape (n,), then ``fun`` must return array_like with shape (n,). Or alternatively it can have shape (n, n_points), then ``fun`` must return array_like with shape (n, n_points) (each column corresponds to a single column in ``y``). The choice between the two options is determined by `vectorized` argument (see below). 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. vectorized : bool Whether `fun` is implemented in a vectorized fashion. support_complex : bool, optional Whether integration in a complex domain should be supported. Generally determined by a derived solver class capabilities. 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 the system's rhs evaluations. njev : int Number of the Jacobian evaluations. nlu : int Number of LU decompositions.