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The need to solve a set of n simultaneous nonlinear equations in n unknowns arises in many areas of science and engineering. The equations can be expressed in the form: [equation] [equation] [equation] It will be assumed that at least one real solution exists and that the functions are continuous and possess continuous first derivatives. These assumptions are often good ones in dealing with equations arising in many physical systems. However, the functions themselves are often long, complex and expensive (in terms of computer time) to evaluate and the value of their derivatives can only be inferred from a finite difference approximation. Therefore, solution methods which keep the number of functional evaluations (i.e. the number of times the f vector needs to be evaluated) to a minimum become very attractive.

A review of quasi-Newton methods in nonlinear equation solving and unconstrained optimization