An efficient two-parametric family with memory for nonlinear equations

Cordero, Alicia; Lotfi, Taher; Bakhtiari, Parisa; Torregrosa, Juan R.

doi:10.1007/s11075-014-9846-8

Cited by 37 publications

(31 citation statements)

References 17 publications

(20 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Let f I ⊆ ℝ → ℝ be a sufficiently differentiable function and let γ n and λ n be the varying parameters in iterative scheme (6) obtained by means of (10). If the initial estimate x 0 is close enough to a simple root α of f x , then, the R-order of the iterative method is at least 10.…”

Section: Theoremmentioning

confidence: 99%

“…More recently, some authors have constructed iterative schemes with memory from optimal procedures of different orders, mainly four (see, e.g., [10][11][12]), eight ( [13][14][15], among others), or even general n-point schemes [16,17]. Some good reviews regarding the acceleration of convergence order by using memory are [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…As λ n is calculated by (10), the third derivative of Hermite interpolating polynomial H 6 x satisfies…”

mentioning

confidence: 99%

See 2 more Smart Citations

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Choubey¹,

Cordero

Jaiswal

et al. 2018

Complexity

View full text Add to dashboard Cite

We construct a new biparametric three-point method with memory to highly improve the computational efficiency of its original partner, without adding functional evaluations. In this way, through different estimations of self-accelerating parameters, we have modified an existing seventh-order method. The parameters have been defined by Hermite interpolating polynomial that allows the accelerating effect. In particular, the R-order of the proposed iterative method with memory is increased from seven to ten. A real multidimensional analysis of the stability of this method with memory is made, in order to study its dependence on the initial estimations. Taking into account that usually iterative methods with memory are more stable than their derivative-free partners and the obtained results in this study, the behavior of this scheme shows to be excellent, but for a small domain. Numerical examples and comparison are also provided, confirming the theoretical results.

show abstract

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Choubey¹,

Cordero

Jaiswal

et al. 2018

Complexity

View full text Add to dashboard Cite

show abstract

“…Specifically, we consider the sixth-order method (JMWM6) introduced by Jovana in [3], sixthorder method (LMWM6) introduced by Lotfi et al in [4], seventh-order method (CMWM7) introduced by Cordero et al in [5], twelfth-order method I (LTMWM12I), II (LTMWM12II), III (LTMWM12III) and IV (LTMWM12IV) introduced by Lotfi and Tavakoli in [6], twelfth-order method I (EMWM12I), II (EMWM12II) and III (EMWM12III) introduced by Eftekhari in [7] and fourteenth-order I (LMWM14I) and II (LMWM14II) introduced by Lotfi et al in [8]. Nowadays, high-order methods are important because numerical applications use high precision in their computations; for this reason numerical tests have been carried out using variable precision arithmetic in MATHEMATICA 8 with 100 significant digits.…”

Section: Application To Non-smooth Equationsmentioning

confidence: 99%

Two Efficient Bi-Parametric Derivative Free With Memory Methods for Finding Simple Roots Nonlinear Equations

Jaiswal¹

2016

JAAM

View full text Add to dashboard Cite

The present paper is devoted to the improvement of the existing fourth-and eighth-order derivative free methods without memory proposed by Cordero et al. (2013). To achieve this goal two parameters are introduced which are calculated with the help of Newton's interpolatory polynomial. It is shown that the R-order convergence of the proposed methods has been increased from 4 to 7 and 8 to 14, respectively without any extra evaluation.Two non-smooth examples are demonstrated to confirm theoretical results. Numerically the modified methods are examined along with comparison to recent existing with memory methods.

show abstract

“…Moreover, they allow us to define easily derivative-free schemes by using standard divided difference. The design of this kind of methods has experimented an important growth in the last years, starting with the technique first appeared in the text of Traub [1] and later developed by Petković et al [2] and used by other authors (see [3,4,5,6] and the references inside). Nevertheless, the understanding of their stability has not been developed, as far as we know.…”

Section: Introductionmentioning

confidence: 99%

A multidimensional dynamical approach to iterative methods with memory

Campos

Cordero

Torregrosa

et al. 2015

Applied Mathematics and Computation

Self Cite

View full text Add to dashboard Cite

A dynamical approach on the dynamics of iterative methods with memory for solving nonlinear equations is made. We have designed new methods with memory from Steffensen' or Traub's schemes, as well as from a parametric family of iterative procedures of third-and fourth-order of convergence. We study the local order of convergence of the new iterative methods with memory.We define each iterative method with memory as a discrete dynamical system and we analyze the stability of the fixed points of its rational operator associated on quadratic polynomials. As far as we know, there is no dynamical study on iterative methods with memory and the techniques of complex dynamics used in schemes without memory are not useful in this context. So, we adapt real multidimensional dynamical tools to afford this task.The dynamical behavior of Secant method and the versions of Steffensen' and Traub's schemes with memory, applied on quadratic polynomials, are analyzed. Different kinds of behavior occur, being in general very stable but pathologic cases as attracting strange fixed points are also found. Finally, a modified parametric family of order four, applied on quadratic polynomials, is also studied, showing the bifurcations diagrams and the appearance of chaos.

show abstract

An efficient two-parametric family with memory for nonlinear equations

Cited by 37 publications

References 17 publications

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Two Efficient Bi-Parametric Derivative Free With Memory Methods for Finding Simple Roots Nonlinear Equations

A multidimensional dynamical approach to iterative methods with memory

Contact Info

Product

Resources

About