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lsoda.py
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lsoda.py
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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 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.
"""
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(LSODA, self).__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(LsodaDenseOutput, self).__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)