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, first_step=None, **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.


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.


Initial time.

y0array_like, shape (n,)

Initial state.


Boundary time - the integration won’t continue beyond it. It also determines the direction of the integration.

first_stepfloat or None, optional

Initial step size. Default is None which means that the algorithm should choose.

max_stepfloat, optional

Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver.

rtol, atolfloat 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.

vectorizedbool, 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)).



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)

L. F. Shampine, M. W. Reichelt, “THE MATLAB ODE SUITE”, SIAM J. SCI. COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.


E. Hairer, G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems”, Sec. III.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.


Number of equations.


Current status of the solver: ‘running’, ‘finished’ or ‘failed’.


Boundary time.


Integration direction: +1 or -1.


Current time.


Current state.


Previous time. None if no steps were made yet.


Size of the last successful step. None if no steps were made yet.


Number of evaluations of the right-hand side.


Number of evaluations of the Jacobian.


Number of LU decompositions.



Compute a local interpolant over the last successful step.


Perform one integration step.