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.

This is a variable order method with the order varying automatically from 1 to 5. The general framework of the BDF algorithm is described in . This class implements a quasi-constant step size as explained in . The error estimation strategy for the constant-step BDF is derived in . An accuracy enhancement using modified formulas (NDF) is also implemented.

Can be applied in the complex domain.

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 :None:None:`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 :None:None:`rtol` controls a relative accuracy (number of correct digits), while atol controls absolute accuracy (number of correct decimal places). To achieve the desired :None:None:`rtol`, set atol to be lower than the lowest value that can be expected from rtol * abs(y) so that :None:None:`rtol` dominates the allowable error. If atol is larger than rtol * abs(y) the number of correct digits is not guaranteed. Conversely, to achieve the desired atol set :None:None:`rtol` such that rtol * 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 :None:None:`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 :None:None:`jac` is not :None:None:`None`. If the Jacobian has only few non-zero elements in each row, providing the sparsity structure will greatly speed up the computations . 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 :None:None:`fun` is implemented in a vectorized fashion. Default is False.

Implicit method based on backward-differentiation formulas.

Examples

See :

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GitHub : /scipy/integrate/_ivp/bdf.py#72
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