Order Divided Differences Research Articles

Extending the Applicability of Two-Step Solvers for Solving Equations

We present a local convergence of two-step solvers for solving nonlinear operator equations under the generalized Lipschitz conditions for the first- and second-order derivatives and for the first order divided differences. In contrast to earlier works, we use our new idea of center average Lipschitz conditions, through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates, and better information on the solution. These extensions require the same effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.

A Family of 6-Point n-Ary Interpolating Subdivision Schemes

We derive three-step algorithm based on divided difference to generate a class of 6-point n-ary interpolating sub-division schemes. In this technique second order divided differences have been calculated at specific position and used to insert new vertices. Interpolating sub-division schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. Polynomial generation and polynomial reproduction are attractive properties of sub-division schemes. Shape preserving properties are also significant tool in sub-division schemes. Further, some significant properties of ternary and quaternary sub-division schemes have been elaborated such as continuity, degree of polynomial generation, polynomial reproduction and approximation order. Furthermore, shape preserving property that is monotonicity is also derived. Moreover, the visual performance of proposed schemes has also been demonstrated through several examples.

A family of higher order derivative free methods for nonlinear systems with local convergence analysis

A wide general class of Ostrowski’s families without memory proposed by Behl et al. (Int J Comput Math 90(2):408–422, 2013) is being extended to solve systems of nonlinear equations. This extension uses multidimensional divided differences of first order. Many more new derivative free iterative families with higher order local convergence are presented. In addition, the proposed iterative family for $$\alpha _1=\mathbb {R}-\{0\}$$ and $$\alpha _2=0$$ are special cases of Grau et al. (J Comput Appl Math 237:363–372, 2013) for iterative schemes of fourth and sixth orders. The computational efficiency is compared with some known methods. It is proved that the proposed methods are equally competent with their existing counter parts. Moreover, we present the local convergence analysis of the proposed family of methods based on Lipschitz constants and hypotheses on the divided difference of order one in the more general settings of a Banach space. We expand this way the applicability of these methods, since we used higher derivatives to show convergence of the method in Sect. 3 although such derivatives do not appear in these methods. Numerical experiments are performed which support the theoretical results.

A new characterization of convexity with respect to Chebyshev systems

The notion of $n$th order convexity in the sense of Hopf and Popoviciu is defined via the nonnegativity of the $(n+1)$st order divided differences of a given real-valued function. In view of the well-known recursive formula for divided differences, the nonnegativity of $(n+1)$st order divided differences is equivalent to the $(n-k-1)$st order convexity of the $k$th order divided differences which provides a characterization of $n$th order convexity. The aim of this paper is to apply the notion of higher-order divided differences in the context of convexity with respect to Chebyshev systems introduced by Karlin in 1968. Using a determinant identity of Sylvester, we then establish a formula for the generalized divided differences which enables us to obtain a new characterization of convexity with respect to Chebyshev systems. Our result generalizes that of W\k{a}sowicz which was obtained in 2006. As an application, we derive a necessary condition for functions which can be written as the difference of two functions convex with respect to a given Chebyshev system.

Traces of the Nevanlinna Class on Discrete Sequences

We show that a discrete sequence $$\Lambda $$ of the unit disk is the union of n interpolating sequences for the Nevanlinna class $$\mathcal {N}$$ if and only if the trace of $$\mathcal {N}$$ on $$\Lambda $$ coincides with the space of functions on $$\Lambda $$ for which the divided differences of order $$n-1$$ are uniformly controlled by a positive harmonic function.

Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2-norm. Therefore, we first prove that the L2-norm of the α-th order (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3-α)/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k+(3-α)/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)-th order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results.

Three Step Kurchatov Method for Nondifferentiable Operators

In this paper we find the order of convergence and semilocal convergence of three step Kurchatov-type method. We also analyze the efficiency index and computational efficiency of this method. The semilocal convergence analysis of method has been established by using recurrence relations under the assumption of first order divided difference operators satisfy Lipschitz condition. The convergence theorem and domain of parameters of the method has also been included. The applicability of the proposed convergence analysis is illustrated by solving some numerical examples.

Local convergence of a relaxed two-step Newton like method with applications

We present a local convergence analysis for a relaxed two-step Newton-like method. We use this method to approximate a solution of a nonlinear equation in a Banach space setting. Hypotheses on the first Frechet derivative and on the center divided-difference of order one are used. In earlier studies such as Amat et al. (Numer Linear Algebra Appl 17:639–653, 2010, Appl Math Lett 25(12):2209–2217, 2012, Appl Math Comput 219(24):11341–11347, 2013, Appl Math Comput 219(15):7954–7963, 2013, Reducing Chaos and bifurcations in Newton-type methods. Abstract and applied analysis. Hindawi Publishing Corporation, Cairo, 2013) these methods are analyzed under hypotheses up to the second Frechet derivative and divided differences of order one. Numerical examples are also provided in this work.

Experimental implementation of MIMO model predictive controller-based second order divided difference filter for nonlinear systems

In this paper, we propose an experimental implementation of a nonlinear predictive controller based on the second order divided difference filter (DDF2) which is a derivate-free estimator. The used state estimator avoids the determination of Jacobian matrices required with the extended Kalman filter (EKF) for an easy implementation with nonlinear systems. In addition, the second order Stirling's interpolation of the estimator allows a more accurate a priori state prediction. A constrained nonlinear predictive controller-based DDF2 is designed to control a multivariable three-tank system. In order to generate the control signals, the controller solves a nonlinear objective function. To overcome the offset output error that appeared due to the inaccuracy of the system model, an internal state model correction scheme is adopted at each iteration. Practical results show the reliability of the proposed method in state estimation, setpoint tracking, and smooth control signal generation. The proposed controller outperforms the model predictive control based on linear state space model derived from the linearisation of the system model around the same setpoint trajectory.

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha -th order (1 le alpha le {k+1}) divided difference of the DG error in the L^2 norm is of order {k + frac{3}{2} - frac{alpha }{2}} when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order {2k + frac{3}{2} - frac{alpha }{2}} in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least ({frac{3}{2}k+1})th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order {k+1} in the L^2 norm for the divided differences of DG errors and thus ({2k+1})th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

New iterative methods for nonlinear equations in R

A parameter based on two-step iterative method combining two known third order methods is proposed for solving nonlinear equations in R. The convergence analysis of the method is established to show its fourth order of convergence. In order to enhance the order of convergence from four to seven, its three-step extension is also developed. The increase in the number of function evaluations is reduced by approximating the involved derivative in the third step by the second order divided differences. A number of numerical examples are worked out to demonstrate the efficacy of both the methods. The performance measures used are the number of iterations and the total number of function evaluations required to get an approximation to the root correct up to fifteen decimal places. The informational index and efficiency index of the proposed methods are computed. On comparison with the existing methods, it is observed that our methods give improved performance in terms of computational speed and efficiency.

Fast and stable contour integration for high order divided differences via elliptic functions

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator values functions. This is a classical problem in numerical mathematics but also arises in new applications such as, e.g., the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule.

Ternary shape-preserving subdivision schemes

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order.

Integer valued polynomials on lower triangular integer matrices

Let \(D\) be an integral domain with quotient field \(K\). In this paper we study the algebra of polynomials in \(K[x]\) which map the set of lower triangular \(n\times n\) matrices with coefficients in \(D\) into itself and show that it coincides with the algebra of polynomials whose divided differences of order \(k\) map \(D^{k+1}\) into \(D\) for every \(k< n\). Using this result we describe the polynomial closure of this set of matrices when \(D\) is the ring of integers in a global field.

Discrete analogs of Taikov’s inequality and recovery of sequences given with an error

We consider the problem of the recovery of the kth order divided difference from a sequence given with an errorwith bounded divided difference of nth order, 0 ≤ k < n. The solution of this problem involves an extremal problem similar to that known in the continuous case as Taikov’s inequality.

Analysing the efficiency of some modifications of the secant method

Some modifications of the secant method for solving nonlinear equations are revisited and the local order of convergence is found in a direct symbolic computation. To do this, a development of the inverse of the first order divided differences of a function of several variables in two points is presented. A generalisation of the efficiency index used in the scalar case to several variables is also analysed in order to use the most competitive algorithm.

On a formula for the nth derivative and its applications

In this paper we obtain a formula which involves the n th derivative of the first order divided difference and a corresponding inequality for functions whose (n + 1) -th derivative belongs to a Lp space. These results are a generalization of the results from [1] and [2]. Finally, some examples are given. ]1] T. H . GRO N WA L L, Problem 331., Amer. Math. Monthly, 20 (1913), 138. [2] T. H . GRO N WA L L, Problem 339., Amer. Math. Monthly, 20 (1913), 196

A technique to choose the most efficient method between secant method and some variants

A few variants of the secant method for solving nonlinear equations are analyzed and studied. In order to compute the local order of convergence of these iterative methods a development of the inverse operator of the first order divided differences of a function of several variables in two points is presented using a direct symbolic computation. The computational efficiency and the approximated computational order of convergence are introduced and computed choosing the most efficient method among the presented ones. Furthermore, we give a technique in order to estimate the computational cost of any iterative method, and this measure allows us to choose the most efficient among them.

Bounds for higher order divided differences of analytic functions on a disk

We estimate the absolute value and real part of the order n divided difference of an analytic function on a disk. Particular attention in these estimates is paid to the question of equality.

Derivative free filtering in hydraulic systems for fault identification

Diagnosis of incipient faults in hydraulic systems is of prime importance due to the performance and reliability demands. This paper outlines the application of derivative free filtering in hydraulic systems for the purpose of real-time fault identification. A flexible experimental setup is constructed in order to simulate different types of faults. The method in this paper deals with internal leakage faults. A detailed non-linear model of the hydraulic actuator experimental setup is developed and validated. Robust control strategies typically hide the presence of faults during incipient stages, making identification of the fault difficult. A partial feedback linearization based robust position control strategy is also presented and faults are identified in the presence of robust control. Faults in the hydraulic systems are modeled as parametric faults and second order divided difference filtering (DDF) is used to estimate the states and the parameters. The efficacy of this estimation algorithm is demonstrated using different fault levels as well as different fault growth profiles. The accuracy and the reliability of the methodology are also demonstrated by identifying faults as small as .01 l/s.