U |e C@sddlZddlmZmZddlmZmZmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZddlmZmZd Zd Zd Zd Zd dZddZddZGdddeZ GdddeZ!dS)N) lu_factorlu_solve)issparse csc_matrixeye)splu) group_columns)validate_max_step validate_tolselect_initial_stepnormEPSnum_jacvalidate_first_stepwarn_extraneous) OdeSolver DenseOutputg? cCs|td|ddddf}td|d}t|d|df}|d||||ddddf<d|d<tj|ddS)z6Compute the matrix for changing the differences array.r Nraxis)nparangezeroscumprod)orderfactorIJMr"U/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/integrate/_ivp/bdf.py compute_Rs $r$cCsHt||}t|d}||}t|j|d|d|d|d<dS)zq|||||| }t||}| dkrnd}n|| }|dk r|dks|t| d|||krq| |7} | |7} |dks|dk r|d|||krd} q|} q| | d| | fS)z5Solve the algebraic system resulting from BDF method.rNFr T)copyrangeNEWTON_MAXITERrallisfiniter )funt_new y_predictcpsiLUsolve_luscaletoldyZ dy_norm_old convergedkfdyZdy_normrater"r"r#solve_bdf_system$s8   r@csJeZdZdZejddddddffdd Zdd Zd d Zd d Z Z S)BDFaImplicit 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), 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. 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. 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