NEWTON DERIVED METHODS FOR NONLINEAR EQUATIONS AND INEQUALITIES

Publisher Summary This chapter describes Newton derived methods for nonlinear equations and inequalities. Newton's method is extended to the solution of systems of equations and inequalities, where the number of variables may be larger than the number of equations and inequalities. These extensions of Newton's method converge quadratically when started from a good initial guess, but just like Newton's method, they may diverge when started from a poor initial guess. The chapter presents an alternative stabilized version of Robinson's algorithm that has a greater rate of convergence and requires less strict assumptions than Huang's algorithm. It also presents an iterated version that is more efficient. The stabilization is accomplished by using an Armijo-type gradient method until a battery of tests indicates that one is close enough to a solution for Robinson's extension of Newton's method to converge.