Efficient Gauss-related quadrature for two classes of logarithmic weight functions

Authors:
James S. Ball;Nelson H. F. Beebe
Affiliations:
University of Utah, Salt Lake City, UT;University of Utah, Salt Lake City, UT
Venue:
ACM Transactions on Mathematical Software (TOMS)
Year:
2007

Citing 4
Cited 2

Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules

ACM Transactions on Mathematical Software (TOMS)
Remark on algorithm 726: ORTHPOL—A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules

ACM Transactions on Mathematical Software (TOMS)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Algorithm 867: QUADLOG—a package of routines for generating Gauss-related quadrature for two classes of logarithmic weight functions

ACM Transactions on Mathematical Software (TOMS)

Algorithm 867: QUADLOG—a package of routines for generating Gauss-related quadrature for two classes of logarithmic weight functions

ACM Transactions on Mathematical Software (TOMS)
Gauss quadrature routines for two classes of logarithmic weight functions

Numerical Algorithms

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Abstract

Integrals with logarithmic singularities are often difficult to evaluate by numerical methods. In this work, a quadrature method is developed that allows the exact evaluation (up to machine accuracy) of integrals of polynomials with two general types of logarithmic weights. The total work for the determination of N nodes and points of the quadrature method is O(N2). Subsequently, integrals can be evaluated with O(N) operations and function evaluations, so the quadrature is efficient. This quadrature method can then be used to generate the nonclassical orthogonal polynomials for weight functions containing logarithms and obtain Gauss and Gauss-related quadratures for these weights. Two algorithms for each of the two types of logarithmic weights that incorporate these methods are given in this paper.