Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective