Optimal fourth and eighth-order iterative methods for non-linear equations

Panday, Sunil; Sharma, Ashok; Thangkhenpau, G.

doi:10.1007/s12190-022-01775-2

Cited by 15 publications

(9 citation statements)

References 21 publications

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“…E s (MPa) [195,000,205,000] f y (MPa) [400,500] f ct (MPa) [25,50] φ(mm) [6,40] Tables 2-4 show the numerical performance of the known Rall and Schröder's schemes and the proposed method (9) in solving Model (10). The stopping criterion set was | f (x k+1 )| < 10 −16 , taking as an initial guess, the value x 0 = 9 4000 .…”

Section: Input Rangementioning

confidence: 99%

“…when the multiplicity is unknown. In recent years, some optimal iterative methods for solving nonlinear problems with multiple roots have appeared in the literature, such as [8][9][10][11], for cases with known multiplicity and [12], for unknown multiplicity. In most cases, the iterative expression is complicated, which increases the computational cost.…”

Section: Introductionmentioning

confidence: 99%

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Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

et al. 2023

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In this paper, an iterative procedure to find the solution of a nonlinear constitutive model for embedded steel reinforcement is introduced. The model presents different multiplicities, where parameters are randomly selected within a solvability region. To achieve this, a class of multipoint fixed-point iterative schemes for single roots is modified to find multiple roots, achieving the fourth order of convergence. Complex discrete dynamics techniques are employed to select the members with the most stable performance. The mechanical problem referred to earlier, as well as some academic problems involving multiple roots, are solved numerically to verify the theoretical analysis, robustness, and applicability of the proposed scheme.

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Section: Input Rangementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

et al. 2023

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“…The composition technique is the preferred approach for building an optimal method, along with the use of various approximations and interpolations to reduce the number of functional evaluations. Various fourth-and eighth order optimal iterative algorithms have been developed, see e.g., Chun and Lee [11], Kung and Traub [26], Liu and Wang [28], Pandey et al [35], Parimala et al [36], Rafiullah and Jabeen [40], Sharma and Arora [45] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Abdullah,

Choubey,

Dara

2024

Comm. Math. Appl.

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In this work, two new iterative methods are proposed for finding simple roots of non-linear equations. The new methods are the modifications of the existing work proposed by Rafiullah and Jabeen (New eighth and sixteenth order iterative methods to solve nonlinear equations, International Journal of Applied and Computational Mathematics 3 (2017), 2467 -2476). The first method obtained is of fifth-order two-step with memory method while the second scheme is three-point eight-order optimal without memory method. Firstly, the Hermite interpolation polynomial is employed to eliminate the first derivative. To maintain order, the conversion to a memory scheme was accomplished by introducing self-accelerated parameters, all without requiring any new function evaluations. Additionally, the Gauss quadrature approach was incorporated for the first derivative, aiming to attain optimal eighth-order convergence. In particular, the efficiency index is increased from 1.4953 to 1.7099 and 1.5157 to 1.6817 for fifth-and eighth-orders respectively. Some real-life applicationbased problems, such as Kepler's equation, an ocean engineering problem, Planck's radiation law, a blood rheology model, and the charge between two parallel plates were presented to validate and demonstrate the superiority of the proposed scheme. Another benefit of the proposed scheme is on the restriction of the Newton's method that f ′ (v) ̸ = 0 can be eliminated close to the root.

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“…Equation ( 2) has a quadratic order of convergence for m ≥ 1. While there exist numerous with and without memory iterative methods for computing simple roots [1][2][3][4][5][6], the task of developing efficient methods for finding multiple roots remains challenging. Despite the availability of several without memory iterative methods for multiple roots (see [7][8][9][10] and the references therein), there is a paucity of methods that utilise multiple points with memory and accelerating parameters for computing multiple roots.…”

Section: Introductionmentioning

confidence: 99%

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

et al. 2023

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The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.

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Optimal fourth and eighth-order iterative methods for non-linear equations

Cited by 15 publications

References 21 publications

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

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