Divided Difference Operators Research Articles

A family of Steffensen type methods with seventh-order convergence

In this paper, based on some known fourth-order Steffensen type methods, we present a family of three-step seventh-order Steffensen type iterative methods for solving nonlinear equations and nonlinear systems. For nonlinear systems, a development of the inverse first-order divided difference operator for multivariable function is applied to prove the order of convergence of the new methods. Numerical experiments with comparison to some existing methods are provided to support the underlying theory.

Semilocal convergence conditions for the secant method, using recurrent functions

Using our new concept of recurrent functions, we present new
 sufficient convergence conditions for the secant method
 to a locally unique solution of a nonlinear equation
 in a Banach space. We combine Lipschitz and center-Lipschitz
 conditions on the divided difference operator to obtain the
 semilocal convergence analysis of the secant method. Our error
 bounds are tighter than earlier ones. Moreover, under our
 convergence hypotheses, we can expand the applicability of the
 secant method in cases not covered before [8],
 [9], [12]-[14], [16], [19]-[21]. Application and examples are also
 provided in this study.

Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

We consider the continuous $q$-Jacobi polynomials $\{P_n^{(\alpha,\beta)}(\cdot|q)\}_{n=0}^{\infty}$, extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the $q$-Sobolev character of the non-standard orthogonality for the corresponding family.

Extended sufficient semilocal convergence for the Secant method

We establish new sufficient convergence conditions for the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new semilocal convergence analysis of the Secant method. Moreover, our sufficient convergence conditions expand the applicability of the Secant method in cases not covered before (Dennis, 1971 [9], Hernández et al., 2005 [8], Laasonen, 1969 [15], Ortega and Rheinboldt, 1970 [11], Potra, 1982 [5], Potra, 1985 [7], Schmidt, 1978 [18], Yamamoto, 1987 [12], Wolfe, 1978 [19]). Numerical examples are also provided in this study.

Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere

We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.

A [formula omitted]-Point interpolation formula with its applications to [formula omitted]-identities

Based on Krattenthaler’s determinantal formula and divided difference operators, we give a 2n-point interpolation formula for a polynomial of degree ≤n in one variable. Several applications of this formula, such as q-identities related to divisor functions, finite forms of the quintuple product identity and a bibasic hypergeometric identity, are discussed. We also give an expansion formula for ∏i=1n(y−uqi−1) by using the supersymmetric complete functions and determinant evaluation.

Characterizations of Continuous and Discrete q-Ultraspherical Polynomials

Abstract We characterize the continuous q-ultraspherical polynomials in terms of the special form of the coefficients in the expansion DqPn(x) in the basis {Pn(x)}, Dq being the Askey-Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the L2 norm of the polynomials. A similar characterization is given for the discrete q-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big q-Jacobi polynomials is given.

ON DIVIDED-DIFFERENCE SEMIGROUPS

The Newtonian divided-difference operators generate the nil-Coxeter algebra and semigroup. A bijective correspondence between the nil-Coxeter semigroup and the symmetric group is used to provide braid-like diagrams for the former, and corresponding Reidemeister-type moves for the relations. Conditions are given for similar relations to hold in a skew group ring. Interesting extensions of the nil-Coxeter semigroup are described and given diagrammatic representations.

Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

We consider the continuous $q$-Jacobi polynomials $\{P_n^{(\alpha,\beta)}(\cdot|q)\}_{n=0}^{\infty}$, extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the $q$-Sobolev character of the non-standard orthogonality for the corresponding family.

Frozen divided difference scheme for solving systems of nonlinear equations

The development of an inverse first-order divided difference operator for functions of several variables, as well as a direct computation of the local order of convergence of an iterative method is presented. A generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.

Braided Differential Structure on Weyl Groups, Quadratic Algebras, and Elliptic Functions

We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type A, we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated elliptic Dunkl operators via the representation given by the generalized divided difference operators.

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637–645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big q-Jacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey–Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.

ULM’S method without derivatives

We consider a version of Ulm’s method obtained when the derivative in its definition is replaced by the divided difference operator. The convergence analysis of the proposed method, carried out under the regular continuity assumption, more general and more flexible than the traditional Lipschitz continuity, has produced the convergence condition, existence and uniqueness radii, and error bounds, all shown to be sharp.

On the Steffensen method under the generalized Lipschitz conditions for the divided difference operator

AbstractConvergence of the Steffensen method for solving nonlinear operator equations in the Banach spaces under the generalized Lipschitz condition for the first–order divided differences is investigated. The conditions and quadratic speed of convergence of this method are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Permutation representations on Schubert varieties

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over ${\Bbb C}$ and over ${\Bbb C}[t_1, t_2,\ldots,t_n]$. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators of Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H * G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H * T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H * G (X; k) is isomorphic to the subalgebra of H * T (X; k) annihilated by the divided difference operators.

Statistical Encounters with Complex B-Splines

Complex B-splines as introduced in Forster et al. (Appl. Comput. Harmon. Anal. 20:281-282, 2006) are an extension of Schoenberg's cardinal splines to include complex orders. We exhibit relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and prove a generalization of the Hermite-Genocchi formula. This generalized Hermite- Genocchi formula then gives rise to a more general class of complex B-splines that allows for some interesting stochastic interpretations.

A Fréchet derivative-free cubically convergent method for set-valued maps

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Frechet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.

The symmetric D ω -semi-classical orthogonal polynomials of class one

We give the system of Laguerre–Freud equations associated with the D ω -semi-classical functionals of class one, where D ω is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given.

On difference equations for orthogonal polynomials on nonuniform lattices1

By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey–Wilson operator , following pioneering work by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261–278, connecting and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients.