Construction of Gaussian Quadrature Formulas for Even Weight Functions

Applicable Analysis and Discrete Mathematics

article

Mohammad Masjed-Jamei¹  Gradimir V. Milovanović² 

[1] K.N.Toosi University of Technology P.O. Box 16315–1618;Serbian Academy of Sciences and Arts, Beograd, Serbia University of Niˇs, Faculty of Sciences and Mathematics

关键词: Pollaczek type weight functions;    Jacobi matrix;    Symmetric weight functions;    Symmetric Gaussian quadrature rules;    Orthogonal polynomials;   

DOI  :  10.2298/AADM1701177M

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

Instead of a quadrature rule of Gaussian type with respect to an even weightfunction on (−a, a) with n nodes, we construct the corresponding Gaussianformula on (0, a2) with only [(n+ 1)/2] nodes. Especially, such a procedure isimportant in the cases of nonclassical weight functions, when the elements ofthe corresponding three-diagonal Jacobi matrix must be constructed numerically. In this manner, the influence of numerical instabilities in the process ofconstruction can be significantly reduced, because the dimension of the Jacobimatrix is halved. We apply this approach to Pollaczek’s type weight functions on (−1, 1), to the weight functions on R which appear in the Abel-Planasummation processes, as well as to a class of weight functions with four freeparameters, which covers the generalized ultraspherical and Hermite weights.Some numerical examples are also included.

【 授权许可】

Unknown