hj vddlZddlmZddlmZmZmZddlm Z m Z Gdde Z Gdd e Z dS) N)ode) validate_tolvalidate_first_stepwarn_extraneous) OdeSolver DenseOutputc LeZdZdZddejddddddf fd ZdZd ZxZ S) LSODAaAdams/BDF method with automatic stiffness detection and switching. This is a wrapper to the Fortran solver from ODEPACK [1]_. It switches automatically between the nonstiff Adams method and the stiff BDF method. The method was originally detailed in [2]_. 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. min_step : float, optional Minimum allowed step size. Default is 0.0, i.e., the step size is not bounded and determined solely by the solver. 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 or callable, optional Jacobian matrix of the right-hand side of the system with respect to ``y``. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``. The function will be called as ``jac(t, y)``. 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. lband, uband : int or None Parameters defining the bandwidth of the Jacobian, i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting these requires your jac routine to return the Jacobian in the packed format: the returned array must have ``n`` columns and ``uband + lband + 1`` rows in which Jacobian diagonals are written. Specifically ``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used in `scipy.linalg.solve_banded` (check for an illustration). These parameters can be also used with ``jac=None`` to reduce the number of Jacobian elements estimated by finite differences. vectorized : bool, optional Whether `fun` is implemented in a vectorized fashion. A vectorized implementation offers no advantages for this solver. 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. nfev : int Number of evaluations of the right-hand side. njev : int Number of evaluations of the Jacobian. References ---------- .. [1] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," IMACS Transactions on Scientific Computation, Vol 1., pp. 55-64, 1983. .. [2] L. Petzold, "Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations", SIAM Journal on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148, 1983. 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