Search

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Abstract: In this paper, the qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation are studied by using qualitative theory of polynomial differential system. The phase portraits of system are given under different parametric conditions. Some exact travelling wave solutions of the Zhiber–Shabat equation are obtained. The results presented in this paper improve the previous results. [Copyright &y& Elsevier]

Abstract: Multivariate Birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far. In this paper, we introduce a special new type of multivariate Birkhoff interpolation and present a Newton paradigm for it. Using the algorithms proposed in this paper, we can construct a Hermite system for any interpolation problem of this type and then obtain a Newton basis for the problem w.r.t. the Hermite system. [Copyright &y& Elsevier]

For ordinary differential equations of the form P ( y , v , v ′ , … , v ( n ) ) = 0 which are polynomial in the variables v , v ′ , … , v ( n ) two new reduction methods to first-order equations are considered. The reduced equations are of the forms v ′ = Q 1 ( y , v 1 ∕ q ) and v ′ = ( Q 2 ( y , v ) ) 1 ∕ q , where Q 1 and Q 2 are two polynomials of degree p in v 1 ∕ q and v , respectively, whose coefficients depend on y . In contrast to most of the known reduction methods of these types, which use either q = 1 or q = 2 , in this paper the values of the positive integers p , q are not predetermined. A procedure to obtain the possible values of the integers for which a reduction of any of these types may exist is provided. As a consequence, new reductions that cannot be obtained by other known methods may be found. The new methods have been applied to obtain some reductions and, consequently, new solutions for three polynomial ordinary differential equations related to well-known equations in mathematical physics: the Kuramoto–Sivashinsky equation, a generalized Benney equation and a 5 th-order KdV equation. Some pieces of computer algebra code, written in Maple and implementing the underlying algorithms to derive the reductions, are also included. [ABSTRACT FROM AUTHOR]

Abstract: In the paper [H.S. Kim, Y.J. Ahn, Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain, J. Comput. Appl. Math. 216 (2008) 14–19], Kim and Ahn proved that the best constrained degree reduction of a polynomial over -dimensional simplex domain in -norm equals the best approximation of weighted Euclidean norm of the Bernstein–Bézier coefficients of the given polynomial. In this paper, we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non-existent when . Furthermore, we provide an assumption to guarantee the existence of solution for the constrained degree reduction. [Copyright &y& Elsevier]

In this paper, by using multivariate divided difference (Rabut, 2001) to approximate the partial derivative and the idea of the superposition (Waldron, 2009), we modify a multiquadric quasi-interpolation operator (Ling, 2004) based on a dimension-splitting technique with the property of linear reproducing to gridded data on multi-dimensional spaces, such that a family of proposed multivariate multiquadric quasi-interpolation operators Φ r + 1 has the property of r + 1 ( r ∈ Z , r ⩾ 0 ) degree polynomial reproducing and converges up to a rate of r + 2 . In addition, the proposed quasi-interpolation operator only demands information of location points rather than the derivatives of the function approximated. Moreover, we give the approximation error of our quasi-interpolation operator. Finally, some numerical experiments are shown to confirm the approximation capacity of our quasi-interpolation operator. [ABSTRACT FROM AUTHOR]

Abstract: Multivariate Birkhoff interpolation is the most complicated polynomial interpolation problem and the theory about it is far from systematic and complete. In this paper we derive an Algorithm B-MB (Birkhoff-Monomial Basis) and prove B-MB giving the minimal interpolation monomial basis w.r.t. the lexicographical order of the multivariate Birkhoff problem. This algorithm is the generalization of Algorithm MB in [L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73–87] which is a well known fast algorithm used to compute the interpolation monomial basis of the Hermite interpolation problem. [Copyright &y& Elsevier]

Abstract: The standard way to solve polynomial eigenvalue problems is to convert the matrix polynomial into a matrix pencil that preserves its spectral information — a process known as linearization. When is palindromic, the eigenvalues, elementary divisors, and minimal indices of have certain symmetries that can be lost when using the classical first and second Frobenius companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials which are palindromic whenever is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family. [Copyright &y& Elsevier]

Abstract: In this paper we derive an approximation property of four-point interpolatory curve subdivision, based on local cubic polynomial fitting. We show that when the scheme is used to generate a limit curve that interpolates given irregularly spaced points, sampled from a curve in any space dimension with a bounded fourth derivative, and the chosen parameterization is chordal, the accuracy is fourth order as the mesh size goes to zero. In contrast, uniform and centripetal parameterizations yield only second order. [Copyright &y& Elsevier]

Abstract: This paper addresses a new and efficient linearization technique to solve mixed 0–1 polynomial problems to achieve a global optimal solution. Given a mixed 0–1 polynomial term , where are binary (0–1) variables and is a continuous variable. Also, can be either a positive or a negative parameter. We transform into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms. For this purpose, we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature. As is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations, extensively. Thus, research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers. In this method, each -auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints. In other words, previous methods were dependent on the number of 0–1 variables and therefore, one auxiliary constraint was considered per 0–1 variable, but this method is completely independent of the number of 0–1 variables and this illustrates the high performance of this method in computation considerations. The analysis of this method illustrates the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: IDR() is now recognized as one of the most effective methods, often superior to other Krylov subspace methods, for large nonsymmetric linear systems of equations. In this paper we propose an improvement upon IDR() by incorporating a higher-order stabilization polynomial into IDR(). The proposed algorithm, named GBi-CGSTAB(), shares desirable features with both IDR() and Bi-CGSTAB(). [ABSTRACT FROM AUTHOR]

Abstract: A non-linear structure preserving matrix method for the computation of a structured low rank approximation of the Sylvester resultant matrix of two inexact polynomials and is considered in this paper. It is shown that considerably improved results are obtained when and are processed prior to the computation of , and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of , which leads to a linear structure preserving matrix method, or they can be incremented during the computation of , which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of and that the assignment of and is important because may be a good structured low rank approximation of , but may be a poor structured low rank approximation of because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of and , are shown. [Copyright &y& Elsevier]

Abstract: Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on -splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section. [Copyright &y& Elsevier]

Abstract: By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant () in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of is applied to partial realization problems of 2-D linear systems. [Copyright &y& Elsevier]

Abstract: Precondition plays a critical role in the numerical methods for large and sparse linear systems. It is also true for nonlinear algebraic systems. In this paper incomplete Gröbner basis (IGB) is proposed as a preconditioner of homotopy methods for polynomial systems of equations, which transforms a deficient system into a system with the same finite solutions, but smaller degree. The reduced system can thus be solved faster. Numerical results show the efficiency of the preconditioner. [Copyright &y& Elsevier]

Abstract: It is well-known that the classical orthogonal polynomials of Jacobi, Bessel, Laguerre and Hermite are solutions of a Sturm–Liouville problem of the type where and are polynomials such that and , and is a constant independent of x. Recently, based on the hypergeometric character of the solutions of this differential equation, W. Koepf and M. Masjed-Jamei [A generic formula for the values at the boundary points of monic classical orthogonal polynomials, J. Comput. Appl. Math. 191 (2006) 98–105] found a generic formula, only in terms of the coefficients of and , for the values of the classical orthogonal polynomials at the singular points of the above differential hypergeometric equation. In this paper, we generalize the mentioned result giving the analogous formulas for both the classical -orthogonal polynomials (of the -Hahn tableau) and the classical -orthogonal polynomials. Both are special cases of the classical -orthogonal polynomials, which are solutions of the hypergeometric-type difference equation where is the difference operator introduced by Hahn, and , and being as above. Our approach is algebraic and it does not require hypergeometric functions. [Copyright &y& Elsevier]

Considered in this paper are two systems of polynomials that are orthogonal systems for two different but related inner product spaces. One of these systems is a special case (λ=1/2) of the symmetric Meixner–Pollaczek polynomial systems, Pn(λ)(x/2,π/2), and it turns out that this system is closely related to a system of orthogonal polynomials in the strip, S={z:-1. Moreover, there are some simple operators that connect the systems with each other. We have designated the special case of the symmetric Meixner–Pollaczek polynomial systems by τn and the latter system on the strip by σn, and we have been able to show that this system is the limiting case of the symmetric Meixner–Pollaczek polynomial systems, Pn(λ)(x/2,π/2) as λ→0. [Copyright &y& Elsevier]

In this paper, two weighted essentially nonoscillatory (ENO) schemes are presented on triangular meshes. By combining quadratic polynomials with weights on the ENO stencil, we construct a scheme with second-order accuracy and another scheme with third-order accuracy. Numerical results show the accuracy and stability of the weighted ENO schemes and resolution for discontinuity. [Copyright &y& Elsevier]

In this paper we investigate the inner products in the space of rational functions and the space of Mu¨ntz polynomials. We prove that the orthogonality of a sequence in one of mentioned spaces can be induced by the orthogonality of the corresponding sequence in another space. Finally, we point to several special cases, i.e., some very different classes of well–known functions we represent on a unique way. [Copyright &y& Elsevier]