import numpy as np from scipy.integrate import ode from .common import validate_tol, validate_first_step, warn_extraneous from .base import OdeSolver, DenseOutput class LSODA(OdeSolver): """Adams/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 time derivative of the state ``y`` at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must return an array of the same shape as ``y``. See `vectorized` for more information. 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` may be called in a vectorized fashion. False (default) is recommended for this solver. If ``vectorized`` is False, `fun` will always be called with ``y`` of shape ``(n,)``, where ``n = len(y0)``. If ``vectorized`` is True, `fun` may be called with ``y`` of shape ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of the returned array is the time derivative of the state corresponding with a column of ``y``). Setting ``vectorized=True`` allows for faster finite difference approximation of the Jacobian by methods 'Radau' and 'BDF', but will result in slower execution for this solver. 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. """ def __init__(self, fun, t0, y0, t_bound, first_step=None, min_step=0.0, max_step=np.inf, rtol=1e-3, atol=1e-6, jac=None, lband=None, uband=None, vectorized=False, **extraneous): warn_extraneous(extraneous) super().__init__(fun, t0, y0, t_bound, vectorized) if first_step is None: first_step = 0 # LSODA value for automatic selection. else: first_step = validate_first_step(first_step, t0, t_bound) first_step *= self.direction if max_step == np.inf: max_step = 0 # LSODA value for infinity. elif max_step <= 0: raise ValueError("`max_step` must be positive.") if min_step < 0: raise ValueError("`min_step` must be nonnegative.") rtol, atol = validate_tol(rtol, atol, self.n) solver = ode(self.fun, jac) solver.set_integrator('lsoda', rtol=rtol, atol=atol, max_step=max_step, min_step=min_step, first_step=first_step, lband=lband, uband=uband) solver.set_initial_value(y0, t0) # Inject t_bound into rwork array as needed for itask=5. solver._integrator.rwork[0] = self.t_bound solver._integrator.call_args[4] = solver._integrator.rwork self._lsoda_solver = solver def _step_impl(self): solver = self._lsoda_solver integrator = solver._integrator # From lsoda.step and lsoda.integrate itask=5 means take a single # step and do not go past t_bound. itask = integrator.call_args[2] integrator.call_args[2] = 5 solver._y, solver.t = integrator.run( solver.f, solver.jac or (lambda: None), solver._y, solver.t, self.t_bound, solver.f_params, solver.jac_params) integrator.call_args[2] = itask if solver.successful(): self.t = solver.t self.y = solver._y # From LSODA Fortran source njev is equal to nlu. self.njev = integrator.iwork[12] self.nlu = integrator.iwork[12] return True, None else: return False, 'Unexpected istate in LSODA.' def _dense_output_impl(self): iwork = self._lsoda_solver._integrator.iwork rwork = self._lsoda_solver._integrator.rwork order = iwork[14] h = rwork[11] yh = np.reshape(rwork[20:20 + (order + 1) * self.n], (self.n, order + 1), order='F').copy() return LsodaDenseOutput(self.t_old, self.t, h, order, yh) class LsodaDenseOutput(DenseOutput): def __init__(self, t_old, t, h, order, yh): super().__init__(t_old, t) self.h = h self.yh = yh self.p = np.arange(order + 1) def _call_impl(self, t): if t.ndim == 0: x = ((t - self.t) / self.h) ** self.p else: x = ((t - self.t) / self.h) ** self.p[:, None] return np.dot(self.yh, x)