package scipy

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type tag = [
  1. | `BDF
]
type t = [ `BDF | `Object ] Obj.t
val of_pyobject : Py.Object.t -> t
val to_pyobject : [> tag ] Obj.t -> Py.Object.t
val create : ?max_step:float -> ?rtol:Py.Object.t -> ?atol:Py.Object.t -> ?jac: [ `Ndarray of [> `Ndarray ] Np.Obj.t | `Callable of Py.Object.t | `Sparse_matrix of Py.Object.t ] -> ?jac_sparsity:[> `ArrayLike ] Np.Obj.t -> ?vectorized:bool -> ?first_step:float -> ?extraneous:(string * Py.Object.t) list -> fun_:Py.Object.t -> t0:float -> y0:[> `Ndarray ] Np.Obj.t -> t_bound:float -> unit -> t

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 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. 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). 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.

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.

val dense_output : [> tag ] Obj.t -> Py.Object.t

Compute a local interpolant over the last successful step.

Returns ------- sol : `DenseOutput` Local interpolant over the last successful step.

val step : [> tag ] Obj.t -> string option

Perform one integration step.

Returns ------- message : string or None Report from the solver. Typically a reason for a failure if `self.status` is 'failed' after the step was taken or None otherwise.

val n : t -> int

Attribute n: get value or raise Not_found if None.

val n_opt : t -> int option

Attribute n: get value as an option.

val status : t -> string

Attribute status: get value or raise Not_found if None.

val status_opt : t -> string option

Attribute status: get value as an option.

val t_bound : t -> float

Attribute t_bound: get value or raise Not_found if None.

val t_bound_opt : t -> float option

Attribute t_bound: get value as an option.

val direction : t -> float

Attribute direction: get value or raise Not_found if None.

val direction_opt : t -> float option

Attribute direction: get value as an option.

val t : t -> float

Attribute t: get value or raise Not_found if None.

val t_opt : t -> float option

Attribute t: get value as an option.

val y : t -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t

Attribute y: get value or raise Not_found if None.

val y_opt : t -> [ `ArrayLike | `Ndarray | `Object ] Np.Obj.t option

Attribute y: get value as an option.

val t_old : t -> float

Attribute t_old: get value or raise Not_found if None.

val t_old_opt : t -> float option

Attribute t_old: get value as an option.

val step_size : t -> float

Attribute step_size: get value or raise Not_found if None.

val step_size_opt : t -> float option

Attribute step_size: get value as an option.

val nfev : t -> int

Attribute nfev: get value or raise Not_found if None.

val nfev_opt : t -> int option

Attribute nfev: get value as an option.

val njev : t -> int

Attribute njev: get value or raise Not_found if None.

val njev_opt : t -> int option

Attribute njev: get value as an option.

val nlu : t -> int

Attribute nlu: get value or raise Not_found if None.

val nlu_opt : t -> int option

Attribute nlu: get value as an option.

val to_string : t -> string

Print the object to a human-readable representation.

val show : t -> string

Print the object to a human-readable representation.

val pp : Format.formatter -> t -> unit

Pretty-print the object to a formatter.

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