Multidimensional parametrization and numerical solution of systems of nonlinear equations

The numerical solution of a system of nonlinear algebraic or transcendental equations is examined within the framework of the parameter continuation method. An earlier result of the author according to which the best parameters should be sought in the tangent space of the solution set of this system is now refined to show that the directions of the eigenvectors of a certain linear self-adjoint operator should be used for finding these parameters. These directions correspond to the extremal values of the quadratic form associated with the above operator. The parametric approximation of curves and surfaces is considered.