Divided Differences Research Articles

Numerical solution of third-order Robin boundary value problems using diagonally multistep block method

This numerical study exclusively focused on developing a diagonally multistep block method of order five (2DDM5) to get the approximate solution of the third-order Robin boundary value problems directly. The mathematical derivation of the developed 2DDM5 method is by approximating the integrand function with Lagrange interpolation polynomial. The proposed direct integrator scheme was applied to compute numerical solution at two-point concurrently. Shooting technique adapted with the Newtons divided difference interpolation method was implemented throughout the proposed algorithm. The theoretical characteristics of the developed method including the order, consistency, zero-stable and convergence are discussed. The method are tested on four Robin boundary value problems. The numerical results signify that the computational performances of the proposed method is competitive in terms of accuracy and efficiency than the existing method.

New properties of the divided difference of psi and polygamma functions

New properties of the divided difference of psi and polygamma functions

The tests of the belonging of functions to the classes Fn(D) and Qn(D)

In the work with the aid of the divided and finite differences are introduced classes Fn(D) and Qn(D) of functions holomorphic in region D. Different properties of functions from these classes are studied, the tests of the belonging of functions to the mentioned classes are established.

Relation Between Extended Stirling Numbers and q-B-splines

Stirling numbers of second kind S(n,k) denotes the number of ways partitioning a set of n elements into k nonempty sets. There are many types of stirling numbers which are studied up to now. In this study, we use extended stirling numbers of second kind which are defined for arbitrary reals. First, we define a relation between extended stirling numbers and q-B-splines by using the property that divided differences have a representation with q-B-splines. In addition, we derive identities on stirling numbers and q-integral of q-B-splines. Furthermore, we give q-generating functions of extended stirling numbers and define a q-difference equation for this function.

Some Remarks on Results Related to ∇-Convex Function

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

Some Remarks on Results Related to ∇-Convex Function

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.

Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

A Family of Fifth and Sixth Convergence Order Methods for Nonlinear Models

We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.

Quantum storage in quantum ferromagnets

We must protect inherently fragile quantum data to unlock the potential of quantum technologies. A pertinent concern in schemes for quantum storage is their potential for near-term implementation. Since Heisenberg ferromagnets are readily available, we investigate their potential for robust quantum storage. We propose to use permutation-invariant quantum codes to store quantum data in Heisenberg ferromagnets, because the ground space of any Heisenberg ferromagnet must be symmetric under any permutation of the underlying qubits. By exploiting an area law on the expected energy of Pauli errors, we show that increasing the effective dimension of Heisenberg ferromagnets can improve the storage lifetime. When the effective dimension of Heisenberg ferromagnets is maximal, we also obtain an upper bound for the storage error. This result relies on perturbation theory, where we use Davis' divided difference representation for Fr\'echet derivatives along with the recursive structure of these divided differences. Our numerical bounds allow us to better understand how quantum memory lifetimes can be enhanced in Heisenberg ferromagnets.

“Smooth rigidity” and Remez-type inequalities

If a \((d+1)\)-smooth function f(x) on \([-1,1]\), with \(\mathrm{max\,}_{[-1,1]}|f(x)|\ge 1,\) has \(d+1\) or more distinct zeroes on \([-1,1]\), then \(\mathrm{max\,}_{[-1,1]}|f^{(d+1)}(x)|\ge 2^{-d-1}(d+1)!\). This follows from the polynomial interpolation of f at its zeroes, with Lagrange’s remainder formula. This is one of the simplest examples of what we call “smooth rigidity”: certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f. In dimensions greater than one, the powerful one-dimension tools such as Lagrange’s remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables (Yomdin in Proc AMS 90(4):538–542, 1984). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of Yomdin (1984), including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a pure multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that the smooth rigidity of a zeroes set Z is approximately the reciprocal Remez constant of Z.

Extended Newton-type Method for Generalized Equations with Hölderian Assumptions

In the present paper, we consider the generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable on a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\mathcal X\to \mathcal Y$ is differentiable at point $\bar{x}$ and linear as well as $\mathcal F$ is a set-valued mapping with closed graph acting between two Banach spaces $\mathcal X$ and $\mathcal Y$. We study the above generalized equation with the help of extended Newton-type method, introduced in [ M. Z. Khaton, M. H. Rashid, M. I. Hossain, Convergence Properties of extended Newton-type Iteration Method for Generalized Equations, Journal of Mathematics Research, 10 (4) (2018), 1--18, DOI:10.5539/jmr.v10n4p1, under the weaker conditions than that are used in Khaton et al. (2018). Indeed, semilocal and local convergence analysis are provided for this method under the conditions that the Frechet derivative of $f$ and the first-order divided difference of $g$ are Hölder continuous on $\Omega$. In particular, we show this method converges superlinearly and these results extend and improve the corresponding results in Argyros (2008) and Khaton $et$ $al.$ (2018).

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

In this article, we present the continuity analysis of the 3D models produced by the tensor product scheme of (m+1)-point binary refinement scheme. We use differences and divided differences of the bivariate refinement scheme to analyze its smoothness. The C^{0}, C^{1} and C^{2} continuity of the general bivariate scheme is analyzed in our approach. This gives us some simple conditions in the form of arithmetic expressions and inequalities. These conditions require the mask and the complexity of the given refinement scheme to analyze its smoothness. Moreover, we perform several experiments by using these conditions on established schemes to verify the correctness of our approach. These experiments show that our results are easy to implement and are applicable for both interpolatory and approximating types of the schemes.

Extended convergence for a fifth‐order novel scheme free from derivatives

The objective in this paper is the expansion of the utilization for a fifth convergence order scheme without derivatives for finding solutions of Banach space valued equations. Conditions of the first order divided difference of the operator involved are only imposed. In this way the use of the scheme is expanded, since in earlier articles the derivatives until order four that do appear in the iterative method are required for setting convergence. Our technique also provides bounds on error distances as well as information about the location of the solutions not given in earlier works. Experiments with concrete problems complete this study.

The Cubature Kalman Filter revisited

In this paper, the construction and effectiveness of the so-called Cubature Kalman Filter (CKF) is revisited, as well as its extensions for higher degrees of precision. In this sense, some stable (with respect to the dimension) cubature rules with a quasi-optimal number of nodes are built, and their numerical performance is checked in comparison with other known formulas. All these cubature rules are suitably placed in the mathematical framework of numerical integration in several variables. A method based on the discretization of higher order partial derivatives by certain divided differences is used to provide stable rules of degrees d=5 and d=7, though it can also be applied for higher dimensions. The application of these old and new formulas to the filter algorithm is tested by means of some examples.

Predictive Analysis and Prognostic Approach of Diabetes Prediction with Machine Learning Techniques

Medical experts indulge in numerous strategies for efficient and predictive measures to model the health status of patients and formulate the patterns that are formed in test results. Most patients would dream of their betterments of their health conditions and thus preventing the progression of any disease. When diabetics is considered in the model, or highly intervening methodology would be required for pre-diabetic individuals. Hidden Markov models have been modified into variant models to derive predictions that accurately produce expected results by investigating patterns of clinical observations from a detailed sample of patient’s dataset. There are yet unanswered and concerning challenges to derive an absolute model for predicting diabetes. The datasets from which the patterns are derived from, still holds levels of in completeness, irregularity and obvious clinical interventions during the diagnosis. The Electronic Medical Records are not furnished with all requisite information in all conditions and scenarios. Due to these irregularities prediction has become highly challenging and there is increase in misclassification rate. Newton’s Divide Difference Method (NDDM) is a conventional model for filling the irregularity in electronic datasets through divided differences. The classical approach considers a polynomial approximation approach, thus leading to Runge Phenomenon. If the interval between data fields id higher, severity of finding the irregularities is even higher. By using this type of technique it helps in improving the accuracy thereby bringing in high level prediction without any error and misclassification. In this technique proposed, a novel approximation technique is implemented using the Euclidean distance parameter over the NDDM approximation to predict the outcomes or risk of Type 2 Diabetes Mellitus among patients. Real world entities in CPCSSN are considered for this study and proposed method is tested. The proposed method filled the irregularity in the data components of EMR with better approximations and the quality of prediction has improved significantly.

Local convergence comparison between frozen Kurchatov and Schmidt–Schwetlick–Kurchatov solvers with applications

In this work we are going to use the Kurchatov–Schmidt–Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by x∗ of operator F. We define F as F:D⊆B1⟶B2 where B1 and B2 stand for Banach spaces, D⊆B1 be a convex set and F be a differentiable mapping according to Fréchet. Under these conditions, for all n=0,1,2,… and 0≤i≤m−1 using Taylor expansion, KSSLS and KLS, when B1=B2 and high order derivatives and divided differences not appearing in these solvers, the results obtained are the restart of the utilization of these iterative solvers. Moreover, we show under the same set of conditions that the local convergence radii are the same, the uniqueness balls coincide but the error estimates on ‖xn−x∗‖ differ. It is worth noticing our results improve the corresponding ones (Grau-Sánchez et al., 2011; Kurchatov, 1971 and Shakno, 2009). Finally, we apply our theoretical results to some numerical examples in order to prove the improvement.

On semilocal convergence of three-step Kurchatov method under weak condition

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the omega condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

On a class of splines free of Gibbs phenomenon

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.

Extended convergence analysis of the Newton–Potra method under weak conditions

We study a nonlinear equation with a nondifferentiable part. The semi-local convergence of the Newton–Potra method is proved under weaker (than in earlier research) conditions on derivatives and divided differences of the first order. Weaker semi-local co

Basin attractors for derivative-free methods to find simple roots of nonlinear equations

Many methods exist for solving nonlinear equations. Several of these methods are derivative-free. One of the oldest is the secant method where the derivative is replaced by a divided difference. Clearly such method will need an additional starting value. Here we consider several derivative-free methods and compare them using the idea of basin of attraction.