Tensor Methods for Nonlinear Equations.

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Abstract
A new class of methods for solving systems of nonlinear equations, called tensor methods, is introduced. Tensor methods are general purpose methods intended especially for problems where the Jacobian matrix at the solution is singular or ill-conditioned. They base each iteration on a quadratic model of the nonlinear function, the standard linear model augmented by a simple second order term. The second order term is selected so that the model interpolates function values from several previous iterations, as well as the current function value and Jacobian. The tensor method requires no more function and derivative information per iteration, and hardly more storage or arithmetic per iteration, than a standard method based on Newton's method. In extensive computational tests, a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model, both on standard nonsingular test problems and on problems where the Jacobian at the solution is singular. (Author)

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