Variational iteration technique for solving a system of nonlinear equations

Ullah

et al. 2017

Preprint

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

Section: Frozen Jacobianmentioning

confidence: 99%

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

Ullah

et al. 2017

Preprint

“…The ideas of preconditioning of system of nonlinear equations are reported by many authors [15][16][17][18]. Let G(x) = [g 1 (x), g 2 (x), .…”

Section: Introductionmentioning

confidence: 99%

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

2017

SeMA

Preconditioning of systems of nonlinear equations modifies the associated Jacobian and provides rapid convergence. The preconditioners are introduced in a way that they do not affect the convergence order of parent iterative method. The multi-step derivativefree iterative method consists of a base method and multi-step part. In the base method, the Jacobian of the system of nonlinear equation is approximated by finite difference operator and preconditioners add an extra term to modify it. The inversion of modified finite difference operator is avoided by computing LU factors. Once we have LU factors, we repeatedly use them to solve lower and upper triangular systems in the multi-step part to enhance the convergence order. The convergence order of m-step Newton iterative method is m + 1. The claimed convergence orders are verified by computing the computational order of convergence and numerical simulations clearly show that the good selection of preconditioning provides numerical stability, accuracy and rapid convergence.Keywords Systems of nonlinear equations · Nonlinear preconditioners · Multi-step iterative methods · Derivative-free Mathematics Subject Classification 65H10 · 65L10 · 65L05 · 65.3 · 65.65 · 80M25 · 80.65

“…Various iterative methods are being developed for finding the simple roots of the nonlinear equation f (x) = 0, by using several different techniques such as Taylor series, quadrature formulas, homotopy and decomposition methods, see [1,2,3,4,5,6,7,8,9,10,11,13,12,14,15]. Some time we come across the nonlinear equations which have zeros of multiplicity m ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

“…Noor and Shah [10,11] also suggested some higher order iterative methods for solving nonlinear equations for finding simple roots and for finding zeros of multiplicity of the nonlinear equations. This technique is recently extended for systems of nonlinear equations [12]. We observe that this technique not only plays an important role for the solution of nonlinear equation for simple roots as well as for multiple roots.…”

Section: Introductionmentioning

confidence: 99%

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

Noor¹,

Shah²

2014

Appl. Math. Inf. Sci.

Self Cite

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.