In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.

In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to 8.8989. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from 1.6818 to 1.7272. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems.

In this paper, we develop new derivative-free four-parametric families of with and without memory iterative methods for determining the roots of nonlinear equations. The family of without memory methods has convergence order eight and supports Kung–Traub's conjecture. It is then extended to obtain the family of with memory methods using the four parameters as accelerating parameters without the need for extra function evaluations. As such, the convergence order increases from 8 to 15.5156 for the family of with memory methods. Analysis of convergence and numerical experiments are carried out on some nonlinear functions to validate the theoretical results and also to demonstrate the effectiveness and applicability of the proposed families of methods.

new efficient families of multipoint iterative methods of fourth and eighth order convergence are constructed for finding simple roots of nonlinear equation Gamma(s) = 0. Both families satisfy the optimality condition of Kung-Traub's conjecture with the family of fourth order methods requiring three function evaluations at each iteration and four evaluations of the functions at each iteration for the family of eighth order methods. Investigation on the theoritical convergence criteria of the families are carried out and fully discussed using the two main theorems which confirm their optimal convergence order. Numerical experiments on test functions are executed by comparing with existing well-known methods of similar nature to demonstrate the effectiveness and good performance of the proposed methods.

New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature.

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.

In this paper, we are presenting an iterative scheme for solving nonlinear equations having multiple roots. The newly developed scheme is an improvement of a method for simple roots and it satisfy the Kung-Traub conjecture, so it is optimal. The weight functional approaches used to develop the method. We have analysed its convergence order and proved it. The methods are numerically compared with known methods in terms of the convergence behaviour of convergence, it shows that the developed schemes are superior to existing methods.

In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods.

In this paper, we propose a new higher order iterative method to find multiple roots of nonlinear equations. The combination of Taylor’s series, Newton’s method and the composition approach are used to derive the new method. It requires three evaluations of the function and two evaluations of the derivative of the function per iteration. The theoretical convergence of the proposed method is proved in the main theorem which establishes sixth order of convergence. We compare the developed method with well-known equivalent existing methods by taking various numerical examples. The numerical results demonstrate the better efficiency of the developed method as compared to some standard iterative methods.

We present in this paper two new families of bi-parametric multipoint higher order iterative methods of optimal order for determining simple roots of the nonlinear equation Ω(s)=0. The proposed families of methods are derivative-free with the optimal three-point fourth order methods requiring only three function evaluations per iteration and the optimal four-point eighth order methods consuming four function evaluations per iteration. Taylor's series expansion and divided difference techniques are employed for the formulation of the methods. Their theoretical convergence properties are thoroughly analysed through the main theorems. Numerical experiments on nonlinear functions with some engineering applications are presented and are compared with some existing methods to demonstrate the effectiveness, applicability and validity of the proposed families of methods. Finally, graphical comparison is made through the basins of attraction which further provide useful information about their dynamical behaviour in the complex plane.

In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

In this work, we propose new fourth and eighth order iterative methods for solving the nonlinear equation f(x)=0 . The proposed methods are of optimal order convergence according to the Kung–Traub’s conjecture requiring three function evaluations per iteration for the fourth order method and four function evaluations per iteration for the eighth order method. The fourth order method uses the composition technique by combining Newton–Steffensen method and Zhou method while the three-step eighth order method uses the Newton–Steffensen–Zhou method with weight functions. The two main theorems fully discuss the convergence criteria of the proposed methods. Numerical experiments are carried out extensively to demonstrate the good performance and effectiveness of our proposed methods by comparing with some existing methods on some test functions. Finally, graphical comparisons by means of basins of attraction are also presented to exhibit their dynamical behaviour in the complex plane.

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new method. Numerous methods have been developed by many researchers whenever the function’s second and higher order derivatives exist in the neighbourhood of the root. Computing the second and higher derivative of a function is a very cumbersome and time consuming task. In terms of low computation cost, the newly proposed method finds the best approximation to the root of non-linear equations by evaluating the function and its first derivative. The proposed method has been theoretically demonstrated to have sixth-order convergence. The proposed method has an efficiency index of 1.56. Several comparisons of the proposed method with the various existing iterative method of the same order have been performed on the number of problems. Finally, the computational results suggest that the newly proposed method is efficient compared to the well-known existing methods.

In this paper, we construct variants of Bawazir’s iterative methods for solving nonlinear equations having simple roots. The proposed methods are two-step and three-step methods, with and without memory. The Newton method, weight function and divided differences are used to develop the optimal fourth- and eighth-order without-memory methods while the methods with memory are derivative-free and use two accelerating parameters to increase the order of convergence without any additional function evaluations. The methods without memory satisfy the Kung–Traub conjecture. The convergence properties of the proposed methods are thoroughly investigated using the main theorems that demonstrate the convergence order. We demonstrate the convergence speed of the introduced methods as compared with existing methods by applying the methods to various nonlinear functions and engineering problems. Numerical comparisons specify that the proposed methods are efficient and give tough competition to some well known existing methods.

In this paper, we have established a new fifth order iterative method for finding multiple roots of nonlinear univariate function with known multiplicity. Many researchers have generated several order techniques, whenever the second and the higher-order derivatives of the function exist in a neighborhood of the root. But the cost of evaluating the second derivative of the function is itself a cumbersome problem. The proposed iteration technique does not require the evaluation of second and higher-order derivatives. We used the weight function approach to derive the proposed technique. The convergence analysis of the proposed method is described exhaustively to reveal the fifth order convergence. Programs are developed in Mathematica 12.2 software to demonstrate the efficacy of the proposed method over the existing method on several numerical test functions. In addition, the presented CPU-time also confirms the improved performance of the proposed methods as compared to some standard iterative methods in the literature.

We have presented a new optimal fourth order iterative method in this paper. Every iteration desires one function evaluation and two first derivative evaluations and therefore the efficiency of this method is 1.5874. Many researchers have generated several order techniques, whenever the second and higher order derivatives of the function exist in a neighbourhood of the root. But the cost of evaluating the second derivative of the function is itself a cumbersome problem. In this paper, we derive a high-order iteration technique to discover the root of nonlinear equations at the low computational cost of the first derivative of the function. In fact, we have obtained the optimal order of convergence which satisfies the Kung and Traub optimality conjecture. Kung and Traub conjectured that the multipoint iteration method without memory based on n-evaluations could achieve optimal convergence order 2 n-1 . The analysis of convergence shows that the new technique has fourth order convergence. In addition, the theoretical convergence property of our technique is fully explored with the help of main theorem that declares the convergence order. Numerical comparisons with some well-known schemes having fourth order of convergence are presented with several examples to verify the efficiency of present technique. These results declare the performance of our technique. Finally, the proposed method is found to be more efficient as compare to some standard iterative methods of same order.

At various stages of fertilization specific spatial and temporal patterns of Ca2+ are required for oocyte maturation. It is crucial to understand the mechanics of Ca2+ regulation in cytosol of oocytes, in order to have better understanding of fertilization process. In this paper, a finite element model of cytosolic calcium regulation in oocyte has been developed for a two-dimensional unsteady state case. The model incorporates the important biophysical processes like diffusion, reaction, leak from endoplasmic recticulum (ER), efflux from cytosol to ER via sarco-ER calcium adenosine triphosphate (SERCA) pumps, buffers and sodium calcium exchanger. Appropriate boundary conditions have been framed. The effect of source, buffer, sodium calcium exchanger, etc. on spatial and temporal patterns of calcium in oocyte have been studied with the help of numerical results.

Calcium dynamics in oocytes plays an important role in oocyte maturation.The calcium concentration is regulated at high levels in oocytes through various mechanisms in order to meet the requirements of oocyte maturation.The understanding of these mechanisms are crucial in understanding the processes of reproduction.In this paper an attempt has been made to develop a finite element model of calcium dynamics in oocyte.The model incorporates the parameters like diffusion coefficient, leak from Endoplasmic Reticulum(ER), and buffers namely 1,2-bis(o-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid(BAPTA) and ethylene glycol-bis(2-aminoethylether)-N,N,N',N '-tetraacetic acid(EGTA).The proposed model is solved numerically using appropriate initial and boundary conditions.A program has been developed in MATLAB 7.11 for the entire problem and simulated on a 32-bit machine to compute the numerical results.The effect of BAPTA, EGTA and Leak from ER is studied in the neighbourhood of L-type calcium channel on calcium distribution in oocyte.