New Mono- and Biaccelerator Iterative Methods with Memory for Nonlinear Equations

Lotfi, Taher; Soleymani, Fazlollah; Shateyi, Stanford; Assari, Paria; Haghani, F. Khaksar

doi:10.1155/2014/705674

Cited by 10 publications

(5 citation statements)

References 14 publications

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“…EI = 16 1∕5 ≃ 1.741, EI = 32 1∕6 ≃ 1.781, EI = 64 1∕7 ≃ 1.814 , respectively. Also, which are better than the other methods given in [1,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], [22,[25][26][27][28][29][30][32][33][34][35][36][37][38][39][40][41], 66]. A comparison between the without memory, with memory and adaptive methods in terms of the maximum efficiency index alongside the number of steps per cycle are given in Fig 5. All algorithms are implemented using symbolic Math of MATHEMATICA [23].…”

Section: Resultsmentioning

confidence: 99%

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Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

Geng¹

2012

Mathematical Sciences

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In this article, we use an effective coupled technique to solve singular integral equations with logarithmic kernel which it occurs in different branches of sciences such as mathematical physics, mechanics, isotropic elastic, etc. At first, we approximate the singular kernel by choosing suitable mesh points and multi-wavelet bases then, Galerkin method is used for solving integral equations. In addition, for simplification and to achieve to a more accuracy, we convert integral equation with logarithmic kernel to integro-differential equation with weakly singular kernel which is solvable with the proposed method. The numerical results show the high accuracy of the solution.

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Section: Resultsmentioning

confidence: 99%

“…Let us describe it a little more. If we use information from the current and only the last iteration, we come up with the method introduced in [34,36]. Also, we have considered…”

Section: Recursive Adaptive Methods With Memorymentioning

confidence: 99%

Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

Geng¹

2012

Mathematical Sciences

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“…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

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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.

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“…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

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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.

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New Mono- and Biaccelerator Iterative Methods with Memory for Nonlinear Equations

Cited by 10 publications

References 14 publications

Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

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

A multidimensional dynamical approach to iterative methods with memory

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