Divided Differences Research Articles

Development of Optimal Iterative Methods with Their Applications and Basins of Attraction

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.

Kurchatov-type methods for non-differentiable Hammerstein-type integral equations

We consider a generic type of nonlinear Hammerstein-type integral equations with the particularity of having non-differentiable kernel of Nemystkii type. So, in order to solve it we consider a uniparametric family of iterative processes derivative free, with the main advantage that for a special value of the involved parameter the iterative method obtained coincides with Newton’s method, that is due to the fact of evaluating the divided difference operator when the two values are the same. We perform a qualitative convergence study by choosing an auxiliary point, that allow us to obtain the existence and separation of solutions of the given equation, that is, local and semilocal convergence balls can be obtained.

Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions

For u>0 with u≠1,2 and m∈N and letfm(x;u)=1u−1ln⁡Γ(x+u)Γ(x+1)−ln⁡(x+u2)−∑k=2mck(u)xk. We prove the complete monotonicity of fm(x;u) for m=2,3 and f4n−1(x;1/2), −f4n−2(x;1/2) for n∈N in x on (0,∞). From these several new sharp bounds for the ratio of gamma functions and divided difference of polygamma functions are established. Lastly, a conjecture is posed.

Improved Robust High-Degree Cubature Kalman Filter Based on Novel Cubature Formula and Maximum Correntropy Criterion with Application to Surface Target Tracking

Robust nonlinear filtering is an important method for tracking maneuvering targets in non-Gaussian noise environments. Although there are many robust filters for nonlinear systems, few of them have ideal performance for mixed Gaussian noise and non-Gaussian noise (such as scattering noise) in practical applications. Therefore, a novel cubature formula and maximum correntropy criterion (MCC)-based robust cubature Kalman filter is proposed. First, the fully symmetric cubature criterion and high-order divided difference are used to construct a new fifth-degree cubature formula using fewer symmetric cubature points. Then, a new cost function is obtained by combining the weighted least-squares method and the MCC loss criterion to deal with the abnormal values of non-Gaussian noise, which enhances the robustness; and statistical linearization methods are used to calculate the approximate result of the measurement process. Thus, the final fifth-degree divided difference–maximum correntropy cubature Kalman filter (DD-MCCKF) framework is constructed. A typical surface-maneuvering target-tracking simulation example is used to verify the tracking accuracy and robustness of the proposed filter. Experimental results indicate that the proposed filter has a higher tracking accuracy and better numerical stability than other common nonlinear filters in non-Gaussian noise environments with fewer cubature points used.

Understanding intergroup conflict: How do children in a divided society perceive group differences?

Understanding intergroup conflict: How do children in a divided society perceive group differences?

Real-time assessment of the Ganga river during pandemic COVID-19 and predictive data modeling by machine learning.

In this study, four water quality parameters were reviewed at 14 stations of river Ganga in pre-, during and post-lockdown and these parameters were modeled by using different machine learning algorithms. Various mathematical models were used for the computation of water quality parameters in pre-, during and post- lockdown period by using Central Pollution Control Board real-time data. Lockdown resulted in the reduction of Biochemical Oxygen Demand ranging from 55 to 92% with increased concentration of dissolved oxygen at few stations. pH was in range of 6.5–8.5 of during lockdown. Total coliform count declined during lockdown period at some stations. The modeling of oxygen saturation deficit showed supremacy of Thomas Mueller model (R2 = 0.75) during lockdown over Streeter Phelps (R2 = 0.57). Polynomial regression and Newton’s Divided Difference model predicted possible values of water quality parameters till 30th June, 2020 and 07th August, 2020, respectively. It was found that predicted and real values were close to each other. Genetic algorithm was used to optimize hyperparameters of algorithms like Support Vector Regression and Radical Basis Function Neural Network, which were then employed for prediction of all examined water quality metrics. Computed values from ANN model were found close to the experimental ones (R2 = 1). Support Vector Regression-Genetic Algorithm Hybrid proved to be very effective for accurate prediction of pH, Biochemical Oxygen Demand, Dissolved Oxygen and Total coliform count during lockdown.Supplementary InformationThe online version contains supplementary material available at 10.1007/s13762-022-04423-1.

The symmetric Dw -Laguerre–Hahn orthogonal polynomial of class one

ABSTRACT We give the system of Laguerre–Freud equations for the recurrence coefficients , , , of orthogonal polynomials with respect to a -Laguerre–Hahn form (linear functional) of class one, where is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases.

Convex-Concave fitting to successively updated data and its application to covid-19 analysis.

Let { n} measurements of a process be provided sequentially, where the process follows a sigmoid shape, but the data have lost sigmoidicity due to measuring errors. If we smooth the data by making least the sum of squares of errors subject to one sign change in the second divided differences, then we obtain a sigmoid approximation. It is known that the optimal fit of this calculation is composed of two separate sections, one best convex and one best concave. We propose a method that starts at the beginning of the data and proceeds systematically to construct the two sections of the fit for the current data, step by step as n is increased. Although the minimization calculation at each step may have many local minima, it can be solved in about {mathcal {O}}(n^2) operations, because of properties of the join between the convex and the concave section. We apply this method to data of daily Covid-19 cases and deaths of Greece, the United States of America and the United Kingdom. These data provide substantial differences in the final approximations. Thus, we evaluate the performance of the method in terms of its capabilities as both constructing a sigmoid-type approximant to the data and a trend detector. Our results clarify the optimization calculation both in a systematic manner and to a good extent. At the same time, they reveal some features of the method to be considered in scenaria that may involve predictions, and as a tool to support policy-making. The results also expose some limitations of the method that may be useful to future research on convex-concave data fitting.

Numerical Processes for Approximating Solutions of Nonlinear Equations

In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article.

Full Hermite Interpolation and Approximation in Topological Fields

By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results.

Extremal functional -interpolation on an arbitrary mesh on the real axis

Abstract The Golomb-de Boor problem of extremal interpolation of infinite real sequences with smallest -norm of the th derivative of the interpolant, , on an arbitrary mesh on the real axis is studied under constraints on the norms of the corresponding divided differences. For this smallest norm, lower estimates are obtained for any in terms of -splines. For the second derivative, this quantity is estimated from below and above by constants depending on the parameter . Bibliography: 13 titles.

An improvement of the Kurchatov method by means of a parametric modification

In this work, a uniparametric generalization of the iterative method due to Kurchatov is presented. The iterative model presented is derivative‐free and approximates the solution of nonlinear equations when the operator is non‐differenciable. As the accessibility of the Kurchatov method is usually a problem in the application of the method, since the set of initial guesses that guarantee the convergence of the method is small, the main objective of this work is to improve the Kurchatov iterative method in its accessibility while maintaining and even increasing its speed of convergence. For this purpose, we introduce a variable parameter in the iterative function of the Kurchatov method that allows us to get a better approximation of the derivative by using a symmetric uniparameteric first‐order divided difference operator. We perform a complex dynamic study that corroborate the improvements in the accessibility region. Moreover, a complete analysis of the local and semilocal convergence is established for the new uniparametric iterative method. Finally, we apply the theoretical results to solve a nonlinear integral equation showing the usefulness of the work.

A Family of Derivative Free Algorithms for Multiple-Roots of Van Der Waals Problem

There are a good number of higher-order iterative methods for computing multiple zeros of nonlinear equations in the available literature. Most of them required first or higher-order derivatives of the involved function. No doubt, high-order derivative-free methods for multiple zeros are more difficult to obtain in comparison with simple zeros and with first order derivatives. This study presents an optimal family of fourth order derivative-free techniques for multiple zeros that requires just three evaluations of function ϕ, per iteration. The approximations of the derivative/s are based on symmetric divided differences. We also demonstrate the application of new algorithms on Van der Waals, Planck law radiation, Manning for isentropic supersonic flow and complex root problems. Numerical results reveal that the proposed derivative-free techniques are more efficient in comparison terms of CPU, residual error, computational order of convergence, number of iterations and the difference between two consecutive iterations with other existing methods.

Divided differences and well-poised q-series

By evaluating divided differences, we present new proofs for some terminating well-poised q-series identities and establish three new formulae for bibasic series, tribasic series and multibasic series.

Mars entry navigation under biases based on adaptive Huber divided difference filter

SummaryTo address the problems of unknown disturbances and measurement outliers at Mars entry phase, three integrated navigation schemes are proposed in this study by complying with divided difference filters, that is, integrated navigation schemes based on the Huber first‐order divided difference filter (HDDF1), adaptive Huber first‐order divided difference filter (AHDDF1), and adaptive Huber second‐order divided difference filter (AHDDF2). The novel versions of predictive state estimation error covariance (PSEEC) and measurement noise covariance (MNC) are deduced by adopting the function of the Huber case to modify the cost function of the standard Kalman filter. To be specific, the HDDF1 is derived by embedding the novel PSEEC and MNC in the general first‐order divided difference filter. To more specifically enhance the performance of the filter under significant biases, an adaptive forgetting factor is introduced in the HDDF1. On that basis, the AHDDF1 is derived. Likewise, the AHDDF2 is yielded. The numerical simulations are presented to verify the effectiveness of the proposed navigation schemes.

Meromorphic Functions of Finite ϕ-order and Linear Askey—Wilson Divided Difference Equations

Meromorphic Functions of Finite ϕ-order and Linear Askey—Wilson Divided Difference Equations

Extremal interpolation with the least value of the norm of the second derivative in

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.

N-Convexity via Delta-Integral Representation of Divided Difference on Time Scales

We introduce the delta-integral representation of divided difference on arbitrary time scales and utilize it to set criteria for n-convex functions involving delta-derivative on time scales. Consequences of the theory appear in terms of estimates which generalize and extend some important facts in mathematical analysis.

Derivative-Free Kurchatov-Type Accelerating Iterative Method for Solving Nonlinear Systems: Dynamics and Applications

Two novel Kurchatov-type first-order divided difference operators were designed, which were used for constructing the variable parameter of three derivative-free iterative methods. The convergence orders of the new derivative-free methods are 3, (5+17)/2≈4.56 and 5. The new derivative-free iterative methods with memory were applied to solve nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) in numerical experiments. The dynamical behavior of our new methods with memory was studied by using dynamical plane. The dynamical planes showed that our methods had good stability.

Improved convergence ball and error analysis of Müller's method

We present an improved convergence analysis of Müller's method for solving nonlinear equation under conditions that the divided differences of order one of the involved function satisfy the Lipschitz conditions. Our result improves the earlier work in literature. Numerical examples are presented to illustrate the theoretical results.