Classe « ode »

Informations générales

Héritage

builtins.object
    ode

Définition

class ode(builtins.object):

help(ode)

A generic interface class to numeric integrators.

Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``.

*Note*: The first two arguments of ``f(t, y, ...)`` are in the
opposite order of the arguments in the system definition function used
by `scipy.integrate.odeint`.

Parameters
----------
f : callable ``f(t, y, *f_args)``
    Right-hand side of the differential equation. t is a scalar,
    ``y.shape == (n,)``.
    ``f_args`` is set by calling ``set_f_params(*args)``.
    `f` should return a scalar, array or list (not a tuple).
jac : callable ``jac(t, y, *jac_args)``, optional
    Jacobian of the right-hand side, ``jac[i,j] = d f[i] / d y[j]``.
    ``jac_args`` is set by calling ``set_jac_params(*args)``.

Attributes
----------
t : float
    Current time.
y : ndarray
    Current variable values.

See also
--------
odeint : an integrator with a simpler interface based on lsoda from ODEPACK
quad : for finding the area under a curve

Notes
-----
Available integrators are listed below. They can be selected using
the `set_integrator` method.

"vode"

    Real-valued Variable-coefficient Ordinary Differential Equation
    solver, with fixed-leading-coefficient implementation. It provides
    implicit Adams method (for non-stiff problems) and a method based on
    backward differentiation formulas (BDF) (for stiff problems).

    Source: http://www.netlib.org/ode/vode.f

    .. warning::

       This integrator is not re-entrant. You cannot have two `ode`
       instances using the "vode" integrator at the same time.

    This integrator accepts the following parameters in `set_integrator`
    method of the `ode` class:

    - atol : float or sequence
      absolute tolerance for solution
    - rtol : float or sequence
      relative tolerance for solution
    - lband : None or int
    - uband : None or int
      Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband.
      Setting these requires your jac routine to return the jacobian
      in packed format, jac_packed[i-j+uband, j] = jac[i,j]. The
      dimension of the matrix must be (lband+uband+1, len(y)).
    - method: 'adams' or 'bdf'
      Which solver to use, Adams (non-stiff) or BDF (stiff)
    - with_jacobian : bool
      This option is only considered when the user has not supplied a
      Jacobian function and has not indicated (by setting either band)
      that the Jacobian is banded. In this case, `with_jacobian` specifies
      whether the iteration method of the ODE solver's correction step is
      chord iteration with an internally generated full Jacobian or
      functional iteration with no Jacobian.
    - nsteps : int
      Maximum number of (internally defined) steps allowed during one
      call to the solver.
    - first_step : float
    - min_step : float
    - max_step : float
      Limits for the step sizes used by the integrator.
    - order : int
      Maximum order used by the integrator,
      order <= 12 for Adams, <= 5 for BDF.

"zvode"

    Complex-valued Variable-coefficient Ordinary Differential Equation
    solver, with fixed-leading-coefficient implementation. It provides
    implicit Adams method (for non-stiff problems) and a method based on
    backward differentiation formulas (BDF) (for stiff problems).

    Source: http://www.netlib.org/ode/zvode.f

    .. warning::

       This integrator is not re-entrant. You cannot have two `ode`
       instances using the "zvode" integrator at the same time.

    This integrator accepts the same parameters in `set_integrator`
    as the "vode" solver.

    .. note::

        When using ZVODE for a stiff system, it should only be used for
        the case in which the function f is analytic, that is, when each f(i)
        is an analytic function of each y(j). Analyticity means that the
        partial derivative df(i)/dy(j) is a unique complex number, and this
        fact is critical in the way ZVODE solves the dense or banded linear
        systems that arise in the stiff case. For a complex stiff ODE system
        in which f is not analytic, ZVODE is likely to have convergence
        failures, and for this problem one should instead use DVODE on the
        equivalent real system (in the real and imaginary parts of y).

"lsoda"

    Real-valued Variable-coefficient Ordinary Differential Equation
    solver, with fixed-leading-coefficient implementation. It provides
    automatic method switching between implicit Adams method (for non-stiff
    problems) and a method based on backward differentiation formulas (BDF)
    (for stiff problems).

    Source: http://www.netlib.org/odepack

    .. warning::

       This integrator is not re-entrant. You cannot have two `ode`
       instances using the "lsoda" integrator at the same time.

    This integrator accepts the following parameters in `set_integrator`
    method of the `ode` class:

    - atol : float or sequence
      absolute tolerance for solution
    - rtol : float or sequence
      relative tolerance for solution
    - lband : None or int
    - uband : None or int
      Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband.
      Setting these requires your jac routine to return the jacobian
      in packed format, jac_packed[i-j+uband, j] = jac[i,j].
    - with_jacobian : bool
      *Not used.*
    - nsteps : int
      Maximum number of (internally defined) steps allowed during one
      call to the solver.
    - first_step : float
    - min_step : float
    - max_step : float
      Limits for the step sizes used by the integrator.
    - max_order_ns : int
      Maximum order used in the nonstiff case (default 12).
    - max_order_s : int
      Maximum order used in the stiff case (default 5).
    - max_hnil : int
      Maximum number of messages reporting too small step size (t + h = t)
      (default 0)
    - ixpr : int
      Whether to generate extra printing at method switches (default False).

"dopri5"

    This is an explicit runge-kutta method of order (4)5 due to Dormand &
    Prince (with stepsize control and dense output).

    Authors:

        E. Hairer and G. Wanner
        Universite de Geneve, Dept. de Mathematiques
        CH-1211 Geneve 24, Switzerland
        e-mail:  ernst.hairer@math.unige.ch, gerhard.wanner@math.unige.ch

    This code is described in [HNW93]_.

    This integrator accepts the following parameters in set_integrator()
    method of the ode class:

    - atol : float or sequence
      absolute tolerance for solution
    - rtol : float or sequence
      relative tolerance for solution
    - nsteps : int
      Maximum number of (internally defined) steps allowed during one
      call to the solver.
    - first_step : float
    - max_step : float
    - safety : float
      Safety factor on new step selection (default 0.9)
    - ifactor : float
    - dfactor : float
      Maximum factor to increase/decrease step size by in one step
    - beta : float
      Beta parameter for stabilised step size control.
    - verbosity : int
      Switch for printing messages (< 0 for no messages).

"dop853"

    This is an explicit runge-kutta method of order 8(5,3) due to Dormand
    & Prince (with stepsize control and dense output).

    Options and references the same as "dopri5".

Examples
--------

A problem to integrate and the corresponding jacobian:

>>> from scipy.integrate import ode
>>>
>>> y0, t0 = [1.0j, 2.0], 0
>>>
>>> def f(t, y, arg1):
...     return [1j*arg1*y[0] + y[1], -arg1*y[1]**2]
>>> def jac(t, y, arg1):
...     return [[1j*arg1, 1], [0, -arg1*2*y[1]]]

The integration:

>>> r = ode(f, jac).set_integrator('zvode', method='bdf')
>>> r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0)
>>> t1 = 10
>>> dt = 1
>>> while r.successful() and r.t < t1:
...     print(r.t+dt, r.integrate(r.t+dt))
1 [-0.71038232+0.23749653j  0.40000271+0.j        ]
2.0 [0.19098503-0.52359246j 0.22222356+0.j        ]
3.0 [0.47153208+0.52701229j 0.15384681+0.j        ]
4.0 [-0.61905937+0.30726255j  0.11764744+0.j        ]
5.0 [0.02340997-0.61418799j 0.09523835+0.j        ]
6.0 [0.58643071+0.339819j 0.08000018+0.j      ]
7.0 [-0.52070105+0.44525141j  0.06896565+0.j        ]
8.0 [-0.15986733-0.61234476j  0.06060616+0.j        ]
9.0 [0.64850462+0.15048982j 0.05405414+0.j        ]
10.0 [-0.38404699+0.56382299j  0.04878055+0.j        ]

References
----------
.. [HNW93] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary
    Differential Equations i. Nonstiff Problems. 2nd edition.
    Springer Series in Computational Mathematics,
    Springer-Verlag (1993)

Constructeur(s)

Signature du constructeur	Description
__init__(self, f, jac=None)

Liste des propriétés

Nom de la propriété	Description
y

Liste des opérateurs

Opérateurs hérités de la classe object

__eq__, __ge__, __gt__, __le__, __lt__, __ne__

Liste des méthodes

Toutes les méthodes Méthodes d'instance Méthodes statiques

Signature de la méthode	Description
get_return_code(self)	Extracts the return code for the integration to enable better control _{[extrait de get_return_code.__doc__]}
integrate(self, t, step=False, relax=False)	Find y=y(t), set y as an initial condition, and return y. _{[extrait de integrate.__doc__]}
set_f_params(self, *args)	Set extra parameters for user-supplied function f. _{[extrait de set_f_params.__doc__]}
set_initial_value(self, y, t=0.0)	Set initial conditions y(t) = y. _{[extrait de set_initial_value.__doc__]}
set_integrator(self, name, **integrator_params)
set_jac_params(self, *args)	Set extra parameters for user-supplied function jac. _{[extrait de set_jac_params.__doc__]}
set_solout(self, solout)
successful(self)	Check if integration was successful. _{[extrait de successful.__doc__]}