class scipy.integrate.BDF(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, jac=None, jac_sparsity=None, vectorized=False, **extraneous)[source]

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.

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.

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 [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 is implemented in a vectorized fashion. Default is False.


[1](1, 2) 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](1, 2, 3) L. F. Shampine, M. W. Reichelt, “THE MATLAB ODE SUITE”, SIAM J. SCI. COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
[3](1, 2) E. Hairer, G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems”, Sec. III.2.
[4](1, 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.
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.


dense_output() Compute a local interpolant over the last successful step.
step() Perform one integration step.