scipy.integrate.BDF¶

class
scipy.integrate.
BDF
(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e06, jac=None, jac_sparsity=None, vectorized=False, first_step=None, **extraneous)[source]¶ Implicit method based on backwarddifferentiation 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 quasiconstant step size as explained in [2]. The error estimation strategy for the constantstep BDF is derived in [3]. An accuracy enhancement using modified formulas (NDF) [2] is also implemented.
Can be applied in the complex domain.
 Parameters
 funcallable
Righthand side of the system. The calling signature is
fun(t, y)
. Heret
is a scalar, and there are two options for the ndarrayy
: It can either have shape (n,); thenfun
must return array_like with shape (n,). Alternatively it can have shape (n, k); thenfun
must return an array_like with shape (n, k), i.e. each column corresponds to a single column iny
. 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). t0float
Initial time.
 y0array_like, shape (n,)
Initial state.
 t_boundfloat
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). 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 1e3 for rtol and 1e6 for atol. jac{None, array_like, sparse_matrix, callable}, optional
Jacobian matrix of the righthand 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 finitedifference approximation.
 jac_sparsity{None, array_like, sparse matrix}, optional
Defines a sparsity structure of the Jacobian matrix for a finitedifference approximation. Its shape must be (n, n). This argument is ignored if jac is not None. If the Jacobian has only few nonzero 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 is implemented in a vectorized fashion. Default is False.
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. 7196, March 1975.
 2(1,2)
L. F. Shampine, M. W. Reichelt, “THE MATLAB ODE SUITE”, SIAM J. SCI. COMPUTE., Vol. 18, No. 1, pp. 122, 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. 117120, 1974.
 Attributes
 nint
Number of equations.
 statusstring
Current status of the solver: ‘running’, ‘finished’ or ‘failed’.
 t_boundfloat
Boundary time.
 directionfloat
Integration direction: +1 or 1.
 tfloat
Current time.
 yndarray
Current state.
 t_oldfloat
Previous time. None if no steps were made yet.
 step_sizefloat
Size of the last successful step. None if no steps were made yet.
 nfevint
Number of evaluations of the righthand side.
 njevint
Number of evaluations of the Jacobian.
 nluint
Number of LU decompositions.
Methods
Compute a local interpolant over the last successful step.
step
()Perform one integration step.