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scipy.integrate.

# ode

class scipy.integrate.ode(f, jac=None)

  A generic interface class to numeric integrators.

  Solve an equation system \(y'(t) = f(t,y)\) 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.

  Methods

  `get_return_code`() | Extracts the return code for the integration to enable better control if the integration fails.
  `integrate`(t[, step, relax]) | Find y=y(t), set y as an initial condition, and return y.
  `set_f_params`(*args) | Set extra parameters for user-supplied function f.
  `set_initial_value`(y[, t]) | Set initial conditions y(t) = y.
  `set_integrator`(name, **integrator_params) | Set integrator by name.
  `set_jac_params`(*args) | Set extra parameters for user-supplied function jac.
  `set_solout`(solout) | Set callable to be called at every successful integration step.
  `successful`() | Check if integration was successful.

  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

  > 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

  > 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).

  > This integrator uses the C translation of the original Fortran 77 ODEPACK library, which can be found at http://www.netlib.org/odepack

  > 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”.

  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)

  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        ]