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We recently switched from the basic Newton-Raphson to a more advanced globally convergent Newton’s method with Line Searches and Backtracking (see Numerical Recipes, Chapter 9.7).

For some special cases many we still need to try too many initial guesses. Does an even more advanced approach exist? What else should we try?

Thanks.

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The short answer is "yes". There is a fairly large community of people who spend their entire lives creating better solutions for such problems.

The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library.

Since you already know about the "Numerical Recipes" book, the next step might be a site like http://plato.asu.edu/sub/zero.html. This lists umpteen packages for solving systems of non-linear equations.

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