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.
- Abramowitz, M. and Stegun, I. A., Eds. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Applied Mathematics Series, vol. 55.) U.S. Department of Commerce, Washington, DC. Google Scholar
- Beebe, N. H. F. and Ball, J. S. 2007. Algorithm 867: QUADLOG---A package of routines for generating Gauss-related quadrature for two classes of logarithmic weight functions. ACM Trans. Math. Softw. 33, 3, (Sep.), 1--20. Google Scholar
- Danloy, B. 1973. Numerical construction of Gaussian quadrature formulas for ∫1 0 (− Log x)ċ xα ċ f(x) ċ dx and ∫&infty; 0 Em(x) ċ f(x) ċ dx. Math. Comput. 27, 124 (Oct.), 861--869.Google Scholar
- Engels, H. 1980. Numerical Quadrature and Cubature. Academic Press, New York, NY, USA.Google Scholar
- Gautschi, W. 1968. Construction of Gauss--Christoffel quadrature formulas. Math. Comput. 22, 102 (Apr.), 251--270.Google Scholar
- Gautschi, W. 1990. Orthogonal polynomials: Theory and practice. In Orthogonal Polynomials: Theory and Practice: Proceedings of the NATO Advanced Study Institute on Orthogonal Polynomials and Their Applications. Ohio State University, Columbus, OH, P. Nevai and M. Ismail, Eds. NATO ASI Series C, Mathematical and Physical Sciences, vol. 294. Kluwer Academic Publishers Group, Norwell, MA, 181--216.Google Scholar
- Gautschi, W. 1994. Algorithm 726: ORTHPOL---A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 1 (Mar.), 21--62. Google Scholar
- Gautschi, W. 1998. Remark on Algorithm 726: ORTHPOL---A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 24, 3 (Sept.), 355--355. Google Scholar
- Krylov, V. I. and Pal'tsev, A. A. 1971. Tables for Numerical Integration of Functions with Logarithmic and Power Singularities. Israel Program for Scientific Translation, Jerusalem, Israel.Google Scholar
- Luke, Y. L. 1977. Algorithms for the Computation of Mathematical Functions. Academic Press, New York, NY.Google Scholar
- Mathews, J. and Walker, R. L. 1970. Mathematical Methods of Physics, 2nd Ed. W. A. Benjamin, Inc., New York, NY.Google Scholar
- Stieltjes, T. J. 1884. Quelques recherches sur la théorie des quadratures dites mécaniques. Ann. Sci. Ec. Norm. Paris, Sér. 3 1, 409--426.Google Scholar
- Wilf, H. S. 1962. Mathematics for the Physical Sciences. John Wiley, New York, NY.Google Scholar
- Zwillinger, D. 1992. Handbook of Integration. Jones and Bartlett Publishers, Inc., One Exeter Plaza, Boston, MA.Google Scholar
Index Terms
- Efficient Gauss-related quadrature for two classes of logarithmic weight functions
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