Solving Systems of Nonlinear Equations by Tensor Methods.

Abstract

Abstract : Tensor methods are a class of general purpose methods for solving systems of nonlinear equations. They are especially intended to efficiently solve problems where the Jacobian matrix at the solution is singular or ill-conditioned, while remaining at least as efficient as standard methods on nonsingular problems. Their distinguishing feature is that they base each iteration on a quadratic model of the nonlinear function. The model has a simple second order term that allows it to interpolate more information about the nonlinear function than standard, linear model based methods, without significantly increasing the cost of forming, storing, or solving the model. This paper summarizes two types of tensor methods, derivative tensor methods that calculate an analytic or finite difference Jacobian at each iteration, and secant tensor methods that avoid Jacobian evaluations. Both are shown to require no more function or derivative information per iteration, and hardly more storage or arithmetic operations per iteration, than standard linear model based methods. Computational results are presented that indicate that both tensor methods are consistently at least as reliable as the corresponding linear model based methods, and are significantly more efficient, both on nonsingular and on singular test problems. (Author)