A multi-step class of iterative methods for nonlinear systems

Soleymani, Fazlollah; Lotfi, Taher; Bakhtiari, Parisa

doi:10.1007/s11590-013-0617-6

Cited by 81 publications

(70 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…On the other hand, a common way to generate new schemes is the direct composition of known methods with a later treatment to reduce the number of functional evaluations (see [13,14,15,16], for example). A variant of this technique is the so called Pseudocomposition, introduced in [17] and [18].…”

Section: Introductionmentioning

confidence: 99%

Solving nonlinear problems by Ostrowski–Chun type parametric families

et al. 2014

View full text Add to dashboard Cite

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence 4 for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the Molecular Interaction Problem.

show abstract

Section: Introductionmentioning

confidence: 99%

Solving nonlinear problems by Ostrowski–Chun type parametric families

et al. 2014

View full text Add to dashboard Cite

show abstract

“…also known as flops-like efficiency index (FEI) [18], wherein C stands for the total computational cost per iteration in terms of the number of functional evaluations along with cost of LU decompositions and solving two triangular systems (based on the flops), to observe the competence of distinctive methods. For solving r linear systems of equations with the frozen LU factorization, the following theorem is generally used to estimate the computational operations [14] Theorem 4.1.…”

Section: Computational Complexitiesmentioning

confidence: 99%

“…In this section, we want to apply our methods to solve four examples, taken from [14,18]. Computations have been carried out using variable precision arithmetic with 600 digits of mantissa in Mathematica 8.…”

Section: Numerical Implementationmentioning

confidence: 99%

See 1 more Smart Citation

Some new efficient multipoint iterative methods for solving nonlinear systems of equations

Lotfi

Bakhtiari

Cordero

et al. 2014

International Journal of Computer Mathematics

Self Cite

View full text Add to dashboard Cite

It is attempted to put forward a new multipoint iterative method of sixth-order convergence for approximating solutions of nonlinear systems of equations. It requires two vector-function and two Jacobian matrices per iteration. Furthermore, we use it as a predictor to derive a general multipoint method. Convergence error analysis, estimating computational complexity, numerical implementation and comparisons are given to verify applicability and validity for the proposed methods.

show abstract

“…For instance, the authors of [8] introduced a class of multi-step iterative methods for solving nonlinear systems of equations which avoid the computation of high order Fréchet derivatives. In summary, multi-step iterative methods are computationally attractive.…”

Section: Introductionmentioning

confidence: 99%

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

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

View full text Add to dashboard Cite

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.

show abstract

A multi-step class of iterative methods for nonlinear systems

Cited by 81 publications

References 16 publications

Solving nonlinear problems by Ostrowski–Chun type parametric families

Solving nonlinear problems by Ostrowski–Chun type parametric families

Some new efficient multipoint iterative methods for solving nonlinear systems of equations

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

Contact Info

Product

Resources

About