scipy.integrate.BDF¶

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.

Parameters:

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 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.

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.

References

[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.

Attributes:

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.

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.

Methods

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

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