In general, a problem that requires the determination of values of the unknowns x1, x2,…, xn for which fi(x1,x2,…,xn) = 0, i = 1,2,…,n where f1, f2,…, fn are given algebraic functions of n variables, i.e. they do not involve derivatives or integrals. This in both theory and practice is a very difficult problem. Such systems of equations arise in many areas, e.g. in numerical methods for nonlinear ordinary and partial differential equations. When n = 1 the single equation can be solved by a variety of effective techniques (all involving iteration); the case of polynomial equations can give rise to complex solutions. For systems of equations, Newton's method and principally its many variants are widely used. For cases of extreme difficulty where, for example, only poor starting approximations are available, methods based on the idea of continuation can be of value.
fi(x1,x2,…,xn) = 0,
i = 1,2,…,n