Numerical solutions of system of nonlinear algebraic equations nonlinear algebraic eq

How about using the classic, the Newton-Raphson :

http://www.library.cornell.edu/nr/bookcpdf/c9-6.pdf [Broken]
http://en.wikipedia.org/wiki/Newton-Raphson_method
http://rkb.home.cern.ch/rkb/AN16pp/node183.html [Broken]

 

Without derivatives the methods are typically less effective (can be 'inefficient' real quick, if it's possible to apply in your case methods utilizing gradients they are typically far more usable & efficient), but ones like the simplex method and conjugate direction methods are zeroth order methods and as such don't use gradients. The simplex method is pretty used for example in unconstrained nonlinear optimization.

http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/unconstrained/nonlinsimplex.html [Broken]

 

For a system of nonlinear algebraic equations, if you want to find all the solutions, you can also consider the continuation method (some people call it homotopy continuation method).
Here is one link:
http://www.math.uic.edu/~jan/PHCpack/phcpack.html