scholarly journals Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions

2005 ◽  
Vol 179 (1-2) ◽  
pp. 263-287 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Aleksandar S. Cvetković
Keyword(s):  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Weight Functions

Applications

2004 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Rational Functions ◽  
Quadrature Rule ◽  
Gauss Quadrature ◽  
Quadrature Rules ◽  
Condition Numbers ◽  
Computational Aspects ◽  
Close Relatives ◽  
Gauss Quadrature Rules

The connection between orthogonal polynomials and quadrature rules has already been elucidated in §1.4. The centerpieces were the Gaussian quadrature rule and its close relatives—the Gauss–Radau and Gauss–Lobatto rules (§1.4.2). There are, however, a number of further extensions of Gauss’s approach to numerical quadrature. Principal among them are Kronrod’s idea of extending an n-point Gauss rule to a (2n + 1)-point rule by inserting n + 1 additional nodes and choosing all weights in such a way as to maximize the degree of exactness (cf. Definition 1.44), and Turán’s extension of the Gauss quadrature rule allowing not only function values, but also derivative values, to appear in the quadrature sum. More recent extensions relate to the concept of accuracy, requiring exactness not only for polynomials of a certain degree, but also for rational functions with prescribed poles. Gauss quadrature can also be adapted to deal with Cauchy principal value integrals, and there are other applications of Gauss’s ideas, for example, in combination with Stieltjes’s procedure or the modified Chebyshev algorithm, to generate polynomials orthogonal on several intervals, or, in comnbination with Lanczos’s algorithm, to estimate matrix functionals. The present section is to discuss these questions in turn, with computational aspects foremost in our mind. We have previously seen in Chapter 2 how Gauss quadrature rules can be effectively employed in the context of computational methods; for example, in computing the absolute and relative condition numbers of moment maps (§2.1.5), or as a means of discretizing measures in the multiple-component discretization method for orthogonal polynomials (§2.2.5) and in Stieltjes-type methods for Sobolev orthogonal polynomials (§2.5.2). It is time now to discuss the actual computation of these, and related, quadrature rules.

scholarly journals An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-21
Author(s):  
Samir A. Ashour ◽  
Hany M. Ahmed
Keyword(s):  
Numerical Evaluation ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

Sign in / Sign up

Export Citation Format

Share Document