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From OpenOpt

Solve system of non-linear equations

subjected to

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• OpenOpt SNLE example
• FuncDesigner examples:
  • Getting all roots of a nonlinear equation
  • basic example of a nonlinear system (with automatic differentiation)
  • getting ALL solutions of a nonlinear system (by interalg)

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in versions prior to 0.34 you should use "NLSP" instead of "SNLE"

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SNLE solvers connected to OpenOpt

Solver	License	Constraints	Can handle derivatives	Info
scipy_fsolve	BSD	None	df	it's a wrapper around MINPACK's hybrd and hybrj algorithms.
interalg	BSD	(since r. 0.36) all; finite box-bounds are required (they may be very huge)	interval analysis is used instead	Guaranteed solution no matter are involved functions convex, nonconvex, multiextremum, ill-conditioned etc (or rather quickly ensures that solution doesn't exist). See the example where interalg works and scipy.optimize fsolve fails. Can obtain ALL solutions of nonlinear equations system in the given box lb_i <= x_i <= ub_i (example).
converter to NLP	BSD	all that NLP solver can handle	all that NLP solver can handle	Example: r = p.solve('nlp:ralg'). It tries to minimize norm(F, 2) till required ftol and contol will be reached.
nssolve	BSD	lb, ub, Aeq, A, c, h (example)	df, dc, dh	This one is primarily for nonsmooth and noisy funcs, uses NSP ralg or amsg2p solvers (former for constrained problems, latter currently for unconstrained) and is intended to be enhanced from time to time, as well as the solvers. ns- can be interpreted as NonSmooth or NoiSy or Naum Shor (Ukrainian academician, my teacher, r-algorithm inventor).

The ones below work very unstable and can't use user-provided gradient, at least for scipy 0.6.0
scipy_anderson, scipy_anderson2, scipy_broyden1, scipy_broyden2, scipy_broyden3, scipy_broyden_generalized
Maybe they will be enhanced in future by someone. See here for details

See also:

SLE (System of linear equations), ODE (ordinary differential equations)