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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.

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. [ABSTRACT FROM AUTHOR]

The dimensionality of parameters and variables is a fundamental issue in physics but is mostly ignored from a mathematical point of view. Difficulties arising from dimensional inconsistency are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the particular case of iterative methods for solving nonlinear equations, dimensionality and scaling affect their robustness: While some classical methods, such as Newton's, are adimensional and scale independent, some other iterations such as Steffensen's are not; their convergence depends on scaling, and their evaluation needs a dimensional congruence. In this paper, we introduce the concept of an adimensional form of a function in order to study the behavior of iterative methods, thus correcting, if possible, some pathological features. From this adimensional form, we will devise an adimensional and scale invariant method based on Steffensen's, which we will call the ASIS method. [ABSTRACT FROM AUTHOR]

Let G be a bounded convex domain with a smooth boundary in which a given system of exponents is not complete. For a class of analytic functions in G that can be represented in G by a series of exponentials, we examine the behavior of coefficients of the series expansion in terms of the growth order near the boundary ∂G. We establish two-sided estimates for the order through characteristics depending only on the indices of the series of exponentials and the support function of the domain (these estimates are strong). As a consequence, we obtain a formula for calculating the growth order through the coefficients. [ABSTRACT FROM AUTHOR]

The primary goal of this work is to introduce two adaptive Steffensen-like methods with memory of the highest efficiency indices. In the existing methods, to improve the convergence order applied to memory concept, the focus has only been on the current and previous iteration. However, it is possible to improve the accelerators, considering the time from the first to the current iterations. Therefore, we achieve superior convergence orders and obtain as high efficiency indices as possible . These are the main contributions of this work. [ABSTRACT FROM AUTHOR]

This paper aims at providing new modifications of Steffensen's scheme for solving nonlinear equations. The new general approach is discussed in detail theoretically. It is observed that a super fast R-order along with an interesting computational efficiency index will be achieved. Dynamical behaviors of the schemes are given to show an enlarged radii of convergence. Some experiments are also brought forward to support the given theory. [ABSTRACT FROM AUTHOR]

Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions.

Based on the fourth-order method of Liu et al. [10], eighth-order three-step iterative methods without memory, which are totally free from derivative calculation and reach the optimal efficiency index are presented. The extension of one of the methods for multiple zeros without the knowledge of multiplicity is presented. Further accelerations will be provided through the concept of with memory iteration methods. Moreover, it is shown by way of illustration that the novel methods are useful on a series of relevant numerical problems when high precision computing is required.

In this work, two-step methods with memory by applying three self-accelerator parameters are proposed and analyzed. In fact, we hit the high bound $7.77200^{\frac {1}{3}}\approx 1.98082$ as the efficiency index. Theoretical results are then supported by numerical examples. [ABSTRACT FROM AUTHOR]

We derive new iterative methods with memory for approximating a simple zero of a nonlinear single variable function. To this end, we first consider several modifications on some existing optimal classes without memory in such a way that their extensions to cases with memory could obtain the higher efficiency index 12 1 4 ≈ 1.86120 . Furthermore, we construct our main method with memory using three self-accelerators. It is demonstrated that this new scheme possesses the very high efficiency index 7.23814 1 3 ≈ 1.93438 . [ABSTRACT FROM AUTHOR]

Some new iterative methods without memory are constructed to reach the optimal convergence order four using only three function evaluations to solve nonlinear equations. Per full cycle, each method from the derived classes is derivative-free. We further extend the derived classes of methods to contribute some schemes with memory, without any additional evaluation of the function. Hence, with a same cost, the contributed methods with memory possess higher R-orders and better efficiency indices. In order to attest the efficiency of the obtained methods, we employ numerical comparisons. [ABSTRACT FROM AUTHOR]

In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory. [ABSTRACT FROM AUTHOR]

Abstract: Based on the fourth-order method of Liu et al. [10], eighth-order three-step iterative methods without memory, which are totally free from derivative calculation and reach the optimal efficiency index are presented. The extension of one of the methods for multiple zeros without the knowledge of multiplicity is presented. Further accelerations will be provided through the concept of with memory iteration methods. Moreover, it is shown by way of illustration that the novel methods are useful on a series of relevant numerical problems when high precision computing is required. [Copyright &y& Elsevier]

Quasi-Newton methods based on least change secant updating formulas that solve linear equations Ax=b in n=dim(x)=dim(b) steps can be expected to solve the smooth nonlinear systems n-step quadratically, i.e. with an r-order of ρ=21/n=1+1/n+O(1/n2). The best rate one can generally expect is ρn-k for some fixed k, where ρn is the positive root of ρn(ρ-1)=1. Irrespective of the shift k, the ratio [image omitted] tends to 1 for large n. To show that this asymptotically optimal rate is actually achieved, one usually has to impose a priori some kind of linear independence condition on the sequence of steps taken by the quasi-Newton iteration in question. Without any such assumptions, we establish in this paper the convergence order ρn for the adjoint Broyden formula proposed by Schlenkrich et al. [S. Schlenkrich, A. Griewank, and A. Walther, Local convergence analysis of TR1 updates for solving nonlinear equations, MATHEON Preprint 337 (2006)]. It requires the evaluation of adjoint vectors, is invariant with respect to linear transformations on the variable domain, and combines the properties of bounded deterioration and heredity. [ABSTRACT FROM AUTHOR]

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods [ABSTRACT FROM AUTHOR]

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Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen's method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions. [ABSTRACT FROM AUTHOR]