package scipy

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, ``jaci,j = d fi / d yj``. ``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, jaci,j != 0 for i-lband <= j <= i+uband. Setting these requires your jac routine to return the jacobian in packed format, jac_packedi-j+uband, j = jaci,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, jaci,j != 0 for i-lband <= j <= i+uband. Setting these requires your jac routine to return the jacobian in packed format, jac_packedi-j+uband, j = jaci,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)

Extracts the return code for the integration to enable better control if the integration fails.

In general, a return code > 0 implies success, while a return code < 0 implies failure.

Notes ----- This section describes possible return codes and their meaning, for available integrators that can be selected by `set_integrator` method.

'vode'

=========== ======= Return Code Message =========== ======= 2 Integration successful. -1 Excess work done on this call. (Perhaps wrong MF.) -2 Excess accuracy requested. (Tolerances too small.) -3 Illegal input detected. (See printed message.) -4 Repeated error test failures. (Check all input.) -5 Repeated convergence failures. (Perhaps bad Jacobian supplied or wrong choice of MF or tolerances.) -6 Error weight became zero during problem. (Solution component i vanished, and ATOL or ATOL(i) = 0.) =========== =======

'zvode'

=========== ======= Return Code Message =========== ======= 2 Integration successful. -1 Excess work done on this call. (Perhaps wrong MF.) -2 Excess accuracy requested. (Tolerances too small.) -3 Illegal input detected. (See printed message.) -4 Repeated error test failures. (Check all input.) -5 Repeated convergence failures. (Perhaps bad Jacobian supplied or wrong choice of MF or tolerances.) -6 Error weight became zero during problem. (Solution component i vanished, and ATOL or ATOL(i) = 0.) =========== =======

'dopri5'

=========== ======= Return Code Message =========== ======= 1 Integration successful. 2 Integration successful (interrupted by solout). -1 Input is not consistent. -2 Larger nsteps is needed. -3 Step size becomes too small. -4 Problem is probably stiff (interrupted). =========== =======

'dop853'

=========== ======= Return Code Message =========== ======= 1 Integration successful. 2 Integration successful (interrupted by solout). -1 Input is not consistent. -2 Larger nsteps is needed. -3 Step size becomes too small. -4 Problem is probably stiff (interrupted). =========== =======

'lsoda'

=========== ======= Return Code Message =========== ======= 2 Integration successful. -1 Excess work done on this call (perhaps wrong Dfun type). -2 Excess accuracy requested (tolerances too small). -3 Illegal input detected (internal error). -4 Repeated error test failures (internal error). -5 Repeated convergence failures (perhaps bad Jacobian or tolerances). -6 Error weight became zero during problem. -7 Internal workspace insufficient to finish (internal error). =========== =======

Find y=y(t), set y as an initial condition, and return y.

Parameters ---------- t : float The endpoint of the integration step. step : bool If True, and if the integrator supports the step method, then perform a single integration step and return. This parameter is provided in order to expose internals of the implementation, and should not be changed from its default value in most cases. relax : bool If True and if the integrator supports the run_relax method, then integrate until t_1 >= t and return. ``relax`` is not referenced if ``step=True``. This parameter is provided in order to expose internals of the implementation, and should not be changed from its default value in most cases.

Returns ------- y : float The integrated value at t

Set extra parameters for user-supplied function f.

Set initial conditions y(t) = y.

Set integrator by name.

Parameters ---------- name : str Name of the integrator. integrator_params Additional parameters for the integrator.

Set extra parameters for user-supplied function jac.

Set callable to be called at every successful integration step.

Parameters ---------- solout : callable ``solout(t, y)`` is called at each internal integrator step, t is a scalar providing the current independent position y is the current soloution ``y.shape == (n,)`` solout should return -1 to stop integration otherwise it should return None or 0

Check if integration was successful.

Attribute t: get value or raise Not_found if None.

Attribute t: get value as an option.

Attribute y: get value or raise Not_found if None.

Attribute y: get value as an option.

Print the object to a human-readable representation.

Print the object to a human-readable representation.

Pretty-print the object to a formatter.