Abstract
This chapter discusses the problem of solving N real equations in N real unknowns. It also presents the best algorithms for the various facets of the problem of solving the given system—getting into a region of local convergence from poor initial estimates; achieving guaranteed convergence to a root from anywhere within a specified region by suitably restricting the functions fi; using a technique in a vicinity of a root (local technique) which is fast, stable and which does not require the user to furnish derivatives of the functions fi; and finally, obtaining additional roots of a nonlinear system without converging again to previously found roots. The chapter mentions two approaches, namely, continuation by differentiation and heuristic search. Both these approaches bring systematically but slowly into a vicinity of a root in which one of the faster techniques is guaranteed to work.
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