A Family of Iterative Schemes for Finding Zeros of Nonlinear Equations having Unknown Multiplicity

Abstract

Abstract:In this paper, we suggest and analyze a new family of iterative methods for finding zeros of multiplicity of nonlinear equations by using the variational iteration technique. These new methods include the Halley method and its variants forms as special cases. We also give several examples to illustrate the efficiency of these methods. Comparison with modified Newton method is also given. These new methods can be considered as an alternative to the modified Newton method. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.

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

“…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.…”

Two new Newton-type methods for the nonlinear equations

“…It is easy to see that the proposed method (1.3) can be reduced to the Halley method [11,16,17,22] if α = 2, β = 2 and γ = −1, and the classical Newton method if α = 1, β = 0 and γ = −1. For accelerating the convergence of the scheme (1.3), we also analyse a new method by introducing the Armijo line search technique.…”

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.…”

“…Noor and his co-researchers [23] have constructed a family of iterative methods for solving nonlinear equations with unknown multiplicity by introducing a preconditioner. They defined a new function…”