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"""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 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). 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 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. """ max_step=np.inf, rtol=1e-3, atol=1e-6, jac=None, lband=None, uband=None, vectorized=False, **extraneous): warn_extraneous(extraneous) super(LSODA, self).__init__(fun, t0, y0, t_bound, vectorized)
if first_step is None: first_step = 0 # LSODA value for automatic selection. elif first_step <= 0: raise ValueError("`first_step` must be positive or None.")
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)
if jac is None: # No lambda as PEP8 insists. def jac(): return None
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
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, 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.'
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)
super(LsodaDenseOutput, self).__init__(t_old, t) self.h = h self.yh = yh self.p = np.arange(order + 1)
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) |