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First-order ODE integrators.
User-friendly interface to various numerical integrators for solving a
system of first order ODEs with prescribed initial conditions::
d y(t)[i]
--------- = f(t,y(t))[i],
d t
y(t=0)[i] = y0[i],
where::
i = 0, ..., len(y0) - 1
class ode
---------
A generic interface class to numeric integrators. It has the following
methods::
integrator = ode(f, jac=None)
integrator = integrator.set_integrator(name, **params)
integrator = integrator.set_initial_value(y0, t0=0.0)
integrator = integrator.set_f_params(*args)
integrator = integrator.set_jac_params(*args)
y1 = integrator.integrate(t1, step=False, relax=False)
flag = integrator.successful()
class complex_ode
-----------------
This class has the same generic interface as ode, except it can handle complex
f, y and Jacobians by transparently translating them into the equivalent
real-valued system. It supports the real-valued solvers (i.e., not zvode) and is
an alternative to ode with the zvode solver, sometimes performing better.
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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)
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