Mathematics of Computation

Author(s): Dirk P. Laurie.
Journal: Math. Comp. 66 (1997), 1133-1145.
MSC (1991): Primary 65D30, 65D32
Retrieve article in: PDF
This article is available free of charge

Abstract: The Jacobi matrix of the $(2n+1)$

-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

1.: D. Boley and G. H. Golub. A survey of matrix inverse eigenvalue problems. Inverse problems, 3:595-622, 1987. MR 89m:65036
2.: F. Caliò, W. Gautschi, and E. Marchetti. On computing Gauss-Kronrod quadrature formulae. Math. Comput., 47:639-650, 1986. MR 88a:65028
3.: F. Caliò and E. Marchetti. Derivation and implementation of an algorithm for singular integrals. Computing, 38:235-245, 1987. MR 88h:65048
4.: S. Elhay and J. Kautsky. Generalized Kronrod Patterson type embedded quadratures. Aplikace Matematiky, 37:81-103, 1992. MR 92j:65026
5.: W. Gautschi and S. Notaris. An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions. Math. Comput., 51: 231-248, 1988. MR 89f:65031
6.: W. Gautschi and S. Notaris. Newton's method and Gauss-Kronrod quadrature. In H. Braß and G. Hämmerlin, editors, Numerical Integration III, pages 60-71, Basel, 1988. Birkhäuser. MR 90m:65104
7.: Walter Gautschi. On generating orthogonal polynomials. SIAM J. Scient. Statist. Comput., 3:289-317, 1982. MR 84e:65022
8.: Walter Gautschi. Questions of numerical condition related to polynomials. In Gene H. Golub, editor, Studies in Numerical Analysis, pages 140-177. Math. Assoc. Amer., 1984. CMP 20:07
9.: Walter Gautschi. Gauss-Kronrod quadrature - a survey. In G. V. Milovanovi\'{c}, editor, Numerical Methods and Approximation Theory III, pages 39-66. University of Ni\v{s}, 1987. MR 89k:41035
10.: Walter Gautschi. Algorithm 726: ORTHPOL - a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software, 20:21-62, 1994.
11.: G.H. Golub and J.H. Welsch. Calculation of Gauss quadrature rules. Math. Comput., 23:221-230, 1969. MR 39:6513
12.: W. B. Gragg and W. J. Harrod. The numerically stable reconstruction of Jacobi matrices from spectral data. Numer. Math., 44:317-335, 1984. MR 85i:65052
13.: D. K. Kahaner and G. Monegato. Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights. Z. Angew. Math. Phys., 29:983-986, 1978. MR 80d:65034
14.: J. Kautsky and S. Elhay. Gauss quadratures and Jacobi matrices for weight functions not of one sign. Math. Comput., 43:543-550, 1984. MR 86f:65049
15.: A. S. Kronrod. Nodes and Weights of Quadrature Formulas. Consultants Bureau, New York, 1965. MR 32:598
16.: D. P. Laurie. Anti-Gaussian quadrature formulas. Math. Comput., 65: 739-747, 1996. MR 96m:65026
17.: G. Monegato. A note on extended Gaussian quadrature rules. Math. Comput., 30:812-817, 1976. MR 55:13746
18.: G. Monegato. Stieltjes polynomials and related quadrature rules. SIAM Review, 24:137-158, 1982. MR 83d:65067
19.: Beresford N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, 1980. MR 81j:65063
20.: Beresford N. Parlett. The new qd algorithms. In Acta Numerica 1995, pages 459-491. Cambridge University Press, 1995. MR 96g:65041
21.: T. N. L. Patterson. The optimum addition of points to quadrature formulae. Math. Comput., 22:847-856, 1968. MR 39:3701
22.: R. Piessens and M. Branders. A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type. Math. Comput., 28:135-139 and 344-347, 1974. MR 49:8293
23.: L. Reichel. Newton interpolation at Leja points. BIT, 30:332-346, 1990. MR 91e:65020
24.: R.A. Sack and A.F. Donovan. An algorithm for Gaussian quadrature given modified moments. Numer. Math., 18:465-478, 1972. MR 46:2829
25.: H. E. Salzer. A recurrence scheme for converting from one orthogonal expansion into another. Comm. ACM., 16:705-707, 1973. MR 52:15956
26.: J.C. Wheeler. Modified moments and Gaussian quadratures. Rocky Mountain J. Math., 4:287-296, 1974. MR 48:12785

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS