Classe « BDF »

Informations générales

Héritage

builtins.object
    OdeSolver
        BDF

Définition

class BDF(OdeSolver):

help(BDF)

Implicit method based on backward-differentiation formulas.

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 [1]_.
This class implements a quasi-constant step size as explained in [2]_.
The error estimation strategy for the constant-step BDF is derived in [3]_.
An accuracy enhancement using modified formulas (NDF) [2]_ is also implemented.

Can be applied in the complex domain.

Parameters
----------
fun : callable
    Right-hand side of the system: the time derivative of the state ``y``
    at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
    scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
    return an array of the same shape as ``y``. See `vectorized` for more
    information.
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), while `atol` controls
    absolute accuracy (number of correct decimal places). To achieve the
    desired `rtol`, set `atol` to be smaller than the smallest value that
    can be expected from ``rtol * abs(y)`` so that `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 `rtol` such that ``rtol * abs(y)`` is always smaller
    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`.
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 [4]_. 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` can be called in a vectorized fashion. Default is False.

    If ``vectorized`` is False, `fun` will always be called with ``y`` of
    shape ``(n,)``, where ``n = len(y0)``.

    If ``vectorized`` is True, `fun` may be called with ``y`` of shape
    ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
    such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
    the returned array is the time derivative of the state corresponding
    with a column of ``y``).

    Setting ``vectorized=True`` allows for faster finite difference
    approximation of the Jacobian by this method, but may result in slower
    execution overall in some circumstances (e.g. small ``len(y0)``).

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] G. D. Byrne, A. C. Hindmarsh, "A Polyalgorithm for the Numerical
       Solution of Ordinary Differential Equations", ACM Transactions on
       Mathematical Software, Vol. 1, No. 1, pp. 71-96, March 1975.
.. [2] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
       COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
.. [3] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations I:
       Nonstiff Problems", Sec. III.2.
.. [4] 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.

Constructeur(s)

Signature du constructeur	Description
__init__(self, fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, jac=None, jac_sparsity=None, vectorized=False, first_step=None, **extraneous)