Matlab Integrators

Non-stiff

• ode45 : Runge-Kutta (4, 5), Dormand-Prince pair | dopri5
• ode23 : Runge-Kutta (2,3) pair of Bogacki and Shampine
• ode113 : variable order Adams-Bashforth-Moulton PECE

Stiff

• ode15s : variable order based on numerical differentation formulas (optionally use BDF backward diff formula)
• ode23s : modified Rosenbrock formula of order 2, one step solver
• ode23t : trapezoidal rule usnig a "free" interpolant
• ode23tb : TR-BDF2, implicit Runge-Kutta (first stage: trapezoidal, second stage: backward diffierentiation of order two)
• ode15i : variable order, fully implicit differential equations

SciPy Integrators

scipy.integrate.odeint uses lsoda which switching between Adams for non-stiff and BDF for stiff. This would be similar to a combination of ode113 and ode15s.

scipy.integrate.ode provides these:

• vode: This is similar to lsoda as it uses a combination of Adams and BDF methods. In fact, the docs sound identical to lsoda.
• lsoda
• dopri5: explicit Runge-Kutta of order (4)5 due to Dormand and Prince. (so same as ode45)
• dopri853: explicit Runge-Kutta 8(5,3) due to Dormand and Prince.

http://people.sc.fsu.edu/~jburkardt/f77_src/odepack/odepack.html

It'd be nice to have LSODI and its variants available for multibody dynamic systems so the mass matrix and forcing vector can be passed in in that linearly implicit form. Looks like scikits.odes includes a wrapper to LSODI: https://pypi.python.org/pypi/scikits.odes