See Task #919

The `scipy.integrate.ode/odeint` API needs some cleanup. This page details a suggestion what a revised API should look like.

== Problems with current api ==

  - `ode` is a class, `odeint` a function
  - `ode` has different semantics than `odeint` for the callback function

== Proposed API ==

The idea would be to follow the lead of the `scipy.interpolate` API (cf. eg. `KroghInterpolate`).

  1. CamelCase classes `Vode`, `Zvode`, `Dopri`, etc., functionality similar to `ode`
  2. Thin wrapper functions `vode`, `zvode`, ..., with functionality similar to `odeint`

The second point could perhaps also be

  2. One wrapper function `odeint` (or something similar) with a suitable keyword argument.

== Use cases ==

=== Integrate a simple ODE ===

{{{
import numpy as np
from scipy.integrate import vode

def f(t,y):
    return -2*y + np.array([np.cos(t),0,0])

t = np.linspace(0, 20, 200)
y = np.vode(f, t=t, y0=[0,1,2])
}}}

== Related ==

  - Related tickets (as of 1 May 2010): #291, #478, #894, #1040, #1169
  - Note about wrapping of LSODAR by Ryan Gutenkunst: http://osdir.com/ml/python.scientific.devel/2005-07/msg00028.html
  - Gabriel Gellner started to work on something related here: https://launchpad.net/pyode
  - Should the design take differential algebraic system in consideration (dae)? An implementation of dae solvers based on current ode implementation is http://scikits.appspot.com/odes
  - There is also a BVP solver in scikits: http://bvp-solver.wikidot.com/,   http://pypi.python.org/pypi/scikits.bvp_solver
  - Another BVP solver: http://pav.iki.fi/software/bvp/index.html

== Open questions ==

  - Are the named classes ``Vode`` etc. discoverable? (Having names of different solvers in class names is obscure.)
    The currently published docs don't show the integrators, e.g. 
    http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html but >>> help(scipy.integrate.ode) does. 
    If ``Vode`` etc. are properly indexed then the sphinx documentation should make them discoverable (with search). And if 
    ode class doesn't shadow the ode module then module docs can be incorporated in the documentation
  - Is complex_ode separate from ode/odeint or should it have the same changes to the API or merged? 
    help(scipy.integrate.complex_ode) doesn't show which integrators are available