Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

Sharma, Ekta; Panday, Sunil; Mittal, Shubham Kumar; Joița, Dan-Marian; Pruteanu, Lavinia Lorena; Jäntschi, Lorentz

doi:10.3390/math11214512

Cited by 4 publications

(5 citation statements)

References 34 publications

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“…Džunić [9] earlier developed an efficient two-parameter method for solving nonlinear equations by using the memory technique and Newton polynomial interpolation to determine the accelerating parameters. For the progress of the memory methods with accelerating parameters in the multi-step iterative schemes, one can refer to [20][21][22][23][24][25][26][27][28][29].…”

Section: Preliminariesmentioning

confidence: 99%

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Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Liu,

Chang,

Kuo

2024

Symmetry

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In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

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Section: Preliminariesmentioning

confidence: 99%

“…where we use Equations ( 27) and (28). Inserting Equation ( 29) into Equation ( 26) and using Equation ( 16), we can obtain…”

Section: Theoremmentioning

confidence: 99%

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Liu,

Chang,

Kuo

2024

Symmetry

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“…Proof. By using the second one in Equation ( 46), subtracting both sides by r and from Equations (38) and (39), we can derive…”

Section: Third New Two-step Iterative Schemementioning

confidence: 99%

“…Traub's technique is a typical method with memory, in which the data that appeared in the previous iteration were adopted in the iteration. For recent progress of the memory method with accelerating parameters in the multistep iterative methods, one can refer to [5,27,[34][35][36][37][38][39][40]. One major goal of this paper is to develop the two-step iterative schemes with a new memory method to determine the accelerating parameters by updating technique with the information at the current step.…”

Section: Introductionmentioning

confidence: 99%

New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index

Liu,

Chang

2024

Mathematics

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In the paper, we iteratively solve a scalar nonlinear equation f(x)=0, where f∈C(I,R),x∈I⊂R, and I includes at least one real root r. Three novel two-step iterative schemes equipped with memory updating methods are developed; they are variants of the fixed-point Newton method. A triple data interpolation is carried out by the two-degree Newton polynomial, which is used to update the values of f′(r) and f′′(r). The relaxation factor in the supplementary variable is accelerated by imposing an extra condition on the interpolant. The new memory method (NMM) can raise the efficiency index (E.I.) significantly. We apply the NMM to five existing fourth-order iterative methods, and the computed order of convergence (COC) and E.I. are evaluated by numerical tests. When the relaxation factor acceleration technique is combined with the modified Dzˇunic´’s memory method, the value of E.I. is much larger than that predicted by the paper [Kung, H.T.; Traub, J.F. J. Assoc. Comput. Machinery 1974, 21]. for the iterative method without memory.

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“…The parameters a 0 and b 0 can be updated to speed up the convergence by the memory-dependent method [2]. One can refer to [3][4][5][6][7][8][9][10] for more memory-dependent iterative methods.…”

Section: Introductionmentioning

confidence: 99%

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

Liu,

Chang

2024

Mathematics

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In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are 6 and 123, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to 163, and a third-degree Newton polynomial is taken to update the values of optimal parameters.

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Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

Cited by 4 publications

References 34 publications

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

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