On the best parametrization

The numerical solution to a system of nonlinear algebraic or transcendental equations with several parameters is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are proved for choosing the best parameters, which provide the best condition number for the system of linear continuation equations. Such parameters have to be sought in the subspace tangent to the solution space of the system of nonlinear equations. This subspace is obtained if the original system of nonlinear equations is solved at the various parameter values from a given set. The parametric approximation of curves and surfaces is considered.