Self-accelerating Parameter Research Articles

Enhanced Ninth-Order Memory-Based Iterative Technique for Efficiently Solving 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.

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

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.

A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter

In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest convergence order of new method is 2 + 6 ≈ 4.4495 . Numerical experiments are made to show the performance of the new method, which supports the theoretical results.

A New Newton Method with Memory for Solving Nonlinear Equations

A new Newton method with memory is proposed by using a variable self-accelerating parameter. Firstly, a modified Newton method without memory with invariant parameter is constructed for solving nonlinear equations. Substituting the invariant parameter of Newton method without memory by a variable self-accelerating parameter, we obtain a novel Newton method with memory. The convergence order of the new Newton method with memory is 1 + 2 . The acceleration of the convergence rate is attained without any additional function evaluations. The main innovation is that the self-accelerating parameter is constructed by a simple way. Numerical experiments show the presented method has faster convergence speed than existing methods.

Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations

In this paper, we present two new derivative-free families of Chebyshev–Halley type methods for solving nonlinear equations numerically. Both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence. Furthermore, accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. The self-accelerating parameter is estimated from the current and previous iteration. This self-accelerating parameter is calculated using Newton’s interpolation polynomial of third and fourth degrees. Consequently, the R-orders of convergence are increased from 4 to 6 and 8 to 12, respectively, without any additional functional evaluation. The results require high-order derivatives reaching up to the eighth derivative. That is why we also present an alternative approach using only the first or at most the fourth derivative. We also obtain the radius of convergence and computable error bounds on the distances involved. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we consider a concrete variety of real life problems coming from different disciplines e.g., Kepler’s equation of motion, Planck’s radiation law problem, fractional conversion in a chemical reactor, the trajectory of an electron in the air gap between two parallel plates, Van der Waal’s equation which explains the behavior of a real gas by introducing in the ideal gas equations, in order to check the applicability and effectiveness of our proposed methods.

A family of Newton-type iterative methods using some special self-accelerating parameters

In this paper, a family of Newton-type iterative methods with memory is obtained for solving nonlinear equations, which uses some special self-accelerating parameters. To this end, we first present two optimal fourth-order iterative methods with memory for solving nonlinear equations. Then we give a novel way to construct the self-accelerating parameter and obtain a family of Newton-type iterative methods with memory. The self-accelerating parameters have the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative methods. The convergence order of the new iterative method is increased from 4 to . Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the new methods. Experiment results show that, compared with the existing methods, the new iterative methods with memory have the advantage of costing less computing time.

A new accelerating technique applied to a variant of Cordero–Torregrosa method

In this paper, a new iterative method with memory is proposed for solving nonlinear equations. The new method is a variant of Cordero–Torregrosa method. More importantly, a new technique is applied to construct the self-accelerating parameter of the new method, which does not increase the computational costs. The new iterative method with memory reaches the order of convergence 2 + 6 ≈ 4 . 4495 . Numerical experiments confirm the theoretical results.

An Ostrowski-type method with memory using a novel self-accelerating parameter

In this paper, an Ostrowski-type method with memory is proposed for solving nonlinear equations. To this end, we first present an optimal fourth-order Ostrowski-type method without memory. Based on this method without memory, an Ostrowski-type method with memory is given by using a simple self-accelerating parameter. The new self-accelerating parameter is constructed by a novel way and has the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative method. The convergence order of the new iterative method is increased from 4 to 2 + 5 ≈ 4.2361 , ( 5 + 13 ) / 2 ≈ 4.30278 and 2 + 6 ≈ 4.4495 , respectively. Numerical experiments are made to show the performance of the new method, which support the theoretical results. From the comparison with some known methods, it is observed that the new method occupies less computing time.

A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2n requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub’s conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

Two Bi-Accelerator Improved with Memory Schemes for Solving Nonlinear Equations

The present paper is devoted to the improvement of theR-order convergence of with memory derivative free methods presented by Lotfi et al. (2014) without doing any new evaluation. To achieve this aim one more self-accelerating parameter is inserted, which is calculated with the help of Newton’s interpolatory polynomial. First theoretically it is proved that theR-order of convergence of the proposed schemes is increased from 6 to 7 and 12 to 14, respectively, without adding any extra evaluation. Smooth as well as nonsmooth examples are discussed to confirm theoretical result and superiority of the proposed schemes.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

A new family of Newton-type iterative methods with and without memory for solving nonlinear equations

In this paper, we present a new family of two-step iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to Kung and Traubs conjecture Kung and Traub (J Appl Comput Math 21:643---651, 1974). Further accelerations of convergence speed are obtained by varying a free parameter in per full iteration. This self-accelerating parameter is calculated by using information available from the current and previous iteration. The corresponding R-order of convergence is increased form 4 to $$\frac{5+\sqrt{17}}{2}\approx 4.5616, \frac{5+\sqrt{21}}{2}\approx 4.7913$$ and 5. The increase of convergence order is attained without any additional calculations so that the family of the methods with memory possesses a very high computational efficiency. Another advantage of the new methods is that they remove the severe condition $$f^{\prime }(x)$$ in a neighborhood of the required root imposed on Newtons method. Numerical comparisons are made to show the performance of our methods, as shown in the illustration examples.

On an efficient family of derivative free three-point methods for solving nonlinear equations

New three-step derivative free families of three-point methods for solving nonlinear equations are presented. First, a new family without memory of optimal order eight, consuming four function evaluations per iteration, is proposed by using two weight functions. The improvement of the convergence rate of this basic family, even up to 50%, is obtained without any additional function evaluation using a self-accelerating parameter. This varying parameter is calculated in each iterative step employing only information from the current and the previous iteration, defining in this way a family with memory. The self-accelerating parameter is calculated applying Newton’s interpolating polynomials of degree scaling from 1 to 4. The corresponding R-orders of convergence are increased from 8 to 10, 11, 6+42≈11.66 and 12, providing very high computational efficiency of the proposed methods with memory. Another convenient fact is that these methods do not use derivatives. Numerical examples and comparison with the existing three-point methods are included to confirm theoretical results.

Three-point methods with and without memory for solving nonlinear equations

A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung–Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The corresponding R-order of convergence is increased from 8 to 4 ( 1 + 5 / 2 ) ≈ 8.472 , 9, 10 and 11. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.