A parameterized multi-step Newton method for solving systems of nonlinear equations

Abstract

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and, therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spacial and temporal dimensions such as, for instance, the Chebyshev pseudospectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radii of convergence than the multi-step Newton method.Keywords: Multi-step iterative methods; Multi-step Newton method; systems of nonlinear equations; partial differential equations; discretization methods for partial differential equations.

A Preconditioned Iterative Method for Solving Systems of Nonlinear Equations Having Unknown Multiplicity

“…Many researchers have proposed iterative methods for solving nonlinear and systems of nonlinear equations for finding simple zeros or zeros with multiplicity greater than one [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The classical iterative method for solving nonlinear and systems of nonlinear equations to find simple zeros is the Newton method, which offers quadratic convergence [16,17] under certain conditions.…”

Multi-step derivative-free preconditioned Newton method for solving systems of nonlinear equations

“…The multi-step Newton method [5] can be written as (2) and its order of convergence is m + 1. Many researchers [6][7][8][9] have proposed higher order multi-step iterative method for solving system of nonlinear equations. In most of real world problems, the closed form expression for the system of nonlinear equations is not always possible.…”

A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations

“…It is of interest that we construct efficient iterative methods for solving system of nonlinear equations. Recently a large community of researchers [2,3,6,23,24,30,31] have contributed in the area of the iterative method for solving system of nonlinear equations associated with partial and ordinary equations. In the efficient class of iterative methods for solving system of nonlinear equations, the multi-step frozen Jacobian iterative methods are good candidates.…”

“…We are interested in constructing frozen Jacobian multi-step iterative methods that offer high convergence order with reasonable computational cost. Recently three frozen Jacobian multi-step iterative methods HJ [19], FTUC [2], MSF [30] for the solving system of nonlinear equations are proposed by different authors. The convergence orders of HJ, FTUC, and MSF, are 2m, 3m − 4 and 3m, respectively.…”