LOCAL CONVERGENCE ANALYSIS OF TENSOR METHODS FOR NONLINEAR EQUATIONS

Tensor methods for nonlinear equations base each iteration upon a stan dard linear model, augmented by a low rank quadratic term that is sele cted in such a way that the mode is efficient to form, store, and solv e. These methods have been shown to be very efficient and robust compu tationally, especially on problems where the Jacobian matrix at the ro ot has a small rank deficiency. This paper analyzes the local converge nce properties of two versions of tensor methods, on problems where th e Jacobian matrix at the root has a null space of rank one. Both metho ds augment the standard linear model by a rank one quadratic term. We show under mild conditions that the sequence of iterates generated by the tensor method based upon an ''ideal'' tensor model converges local ly and two-step Q-superlinearly to the solution with Q-order 3/2, and that the sequence of iterates generated by the tensor method based upo n a practial tensor model converges locally and three-step Q-superline arly to the solution with Q-order 3/2. In the same situation, it is kn own that standard methods converge linearly with constant converging t o 1/2. Hence, tensor methods have theoretical advantages over standard methods. Our analysis also confirms that tensor methods converge at l east quadratically on problems where the Jacobian matrix at the root i s nonsingular.