Multi-Step Preconditioned Newton Methods for Solving Systems of Nonlinear Equations

Abstract

Abstract:The study of different forms of preconditioners for solving a system of nonlinear equations, by using Newton's method, is presented. The preconditioners provide numerical stability and rapid convergence with reasonable computation cost, whenever chosen accurately. Different families of iterative methods can be constructed by using a different kind of preconditioners. The multi-step iterative method consists of a base method and multi-step part. The convergence order of base method is quadratic and each multi-step add an additive factor of one in the previously achieved convergence order. Hence the convergence of order of an m-step iterative method is m + 1. Numerical simulations confirm the claimed convergence order by calculating the computational order of convergence. Finally, the numerical results clearly show the benefit of preconditioning for solving system of nonlinear equations.Keywords: systems of nonlinear equations; nonlinear preconditioners; multi-step iterative methods; Frozen Jacobian When computing simple roots of a system of nonlinear equations, Newton's method [1-4] is a classical, well studied procedure that offers quadratic convergence, under suitably mild regularity assumptions. Many researchers have proposed higher order efficient iterative method [5][6][7][8][9][10][11][12] for solving system of nonlinear equations. Recently, some authors have constructed multi-step iterative methods [13][14][15] for solving system of nonlinear equations. The main benefit of multi-step iterative methods is hidden in the multi-step part. Because, the Jacobian factorization information from the base method is utilized in the multi-step part repeatedly to enhance the convergence order at the cost of the solution of lower and upper triangular systems and a single evaluation of the system of nonlinear equations. X. Wu [16] wrote a note on the improvement of Newton's method for systems of nonlinear equations. In his note, the author introduced the idea of nonlinear preconditioners and showed that the improved Newton method enjoyed the quadratic convergence. Jose et al. [17] used the idea of nonlinear preconditioning to improve the Newton method, for solving the system of nonlinear equations with known multiplicities. Aslam et al. [18] proposed iterative methods for solving nonlinear equations with unknown multiplicity with the help of nonlinear preconditioners. In the another article Aslam and his co-researcher [19] proposed a preconditioned double Newton method with quartic convergence order for the solving system of nonlinear equations. What they have proposed is the following. Let

Multistep Quadrature Based Methods for Nonlinear System of Equations with Singular Jacobian

“…In [8], similar perturbation term was introduced at every step of the corrector factor in the three step frozen Jocobian iterative method for approximating the solution of (1). Other articles like [32] [33] have also developed several iterative methods for Journal of Applied Mathematics and Physics solving (1) with the help of same perturbation term introduced to the target Jacobian of (1). It is worth of note that the diagonal matrix used as the perturbation term in literature has not been significantly modified since introduced in [30].…”

“…Also, its application has not been extended to quadrature based iterative methods in literature. Motivated and inspired by the work in [30] [31] [32] and [33], to develop families of multi-step quadrature based iterative methods with infused perturbation term to its Jacobian in this paper. It is important to note that the perturbation term developed and used in this work is different from the diagonal matrix that is formed by the coordinates functions of the target NLSE (1) used in literature.…”