A note on generalized averaged Gaussian formulas for a class of weight functions
- PDF / 703,866 Bytes
- 17 Pages / 439.642 x 666.49 pts Page_size
- 108 Downloads / 209 Views
A note on generalized averaged Gaussian formulas for a class of weight functions Miodrag M. Spalevi´c1 Received: 13 July 2019 / Accepted: 6 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In the recent paper Notaris (Numer. Math., 142:129–147, 2019) it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss–Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see Spalevi´c (Math. Comp., 76:1483–1492, 2007). Moreover, as almost immediate consequence of our results from Spalevi´c (Math. Comp.,76:1483–1492, 2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129–147, 2019) in a different manner, by means of the Jacobi tridiagonal matrix approach. Keywords Averaged Gaussian quadrature · Gauss–Kronrod quadrature · Stieltjes polynomials Mathematics Subject Classification (2010) Primary 65D32; Secondary 33C45
1 Introduction and preliminary The results of this note are a supplement to the ones in [19]. They are based on the S ones on optimal generalized averaged Gaussian rule G 2n+1 and its numerical construction, considered in [26]. We presented there the effective and simple numerical S 2n+1 , i.e., for constructing both formulas, G method for constructing G 2n+1 and the L generalized averaged Gaussian rule G2n+1 introduced by Laurie in [15]. In the cases 2n+1 , it is recommendwhen Gauss–Kronrod quadrature rule H2n+1 coincides with G 2n+1 by using the method from [26]. By using the useful able to construct H2n+1 as G Miodrag M. Spalevi´c
[email protected] 1
Faculty of Mechanical Engineering, Department of Mathematics, University of Beograd, Kraljice Marije 16, 11120 Belgrade 35, Serbia
Numerical Algorithms
2n+1 via the corresponding Jacobi matrices, we are in the posirepresentation of G tion to prove some facts in this theory in a simple way; see for example [28]. As our approach is not mentioned in [19], we present here in part its influence on the theory of generalized averaged Gaussian and Gauss–Kronrod rules. So, on the basis of this theory, we prove the main results in [19] in a different manner, by means of the Jacobi tridiagonal matrix approach. New formulations of some results are given by S , G L , and by the corresponding monic polynomials F S , F L , the rules G 2n+1 n+1 2n+1 n+1 respectively. In order to explain in more details (for the sake of self containedness), our approach and the obtained results in this theory, derived so far, we split the introduction and preliminary results of the theory of generalized averaged Gaussian rules and their construction in three subsections. 1.1 Gauss quadratures and their Kronrod extensions Let dσ be a given nonnegative measure on a bounded or an unbounded interval [a, b] = supp (dσ ). If σ is an absolutely c
Data Loading...
