Gauss quadratures and Jacobi matrices for weight functions not of one sign
Authors:
J. Kautsky and S. Elhay
Journal:
Math. Comp. 43 (1984), 543550
MSC:
Primary 65D32; Secondary 41A55, 65F15
MathSciNet review:
758201
Fulltext PDF Free Access
Abstract 
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Abstract: Construction of Gauss quadratures with prescribed knots via Jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign. An algorithm is described and is demonstrated by calculating the knots of Kronrod schemes and other Gauss quadratures with prescribed knots.
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 W. Gautschi, "An algorithmic implementation of the generalized Christoffel theorem," in Numerical Integration, ISNM 57 (G. Hämmerlin, ed.), Birkhàuser, Basel, 1982, pp. 89106.
 [2]
 G. H. Golub & J. Kautsky, "Calculation of Gauss quadratures with multiple free and fixed knots," Numer. Math., v. 41, 1983, pp. 147163. MR 703119 (84i:65030)
 [3]
 G. H. Golub & J. H. Welsch, "Calculation of Gauss quadrature rules," Math. Comp., v. 23, 1969, pp. 221230. MR 0245201 (39:6513)
 [4]
 J. Kautsky & S. Elhay, Gauss Quadratures and Jacobi Matrices for Weight Functions Not of One Sign, Report, School of Math. Sci., Flinders University of South Australia, 1983. MR 758201 (86f:65049)
 [5]
 J. Kautsky & G. H. Golub, "Evaluation of Jacobi matrices," Linear Algebra Appl., v. 52/53, 1983, pp. 439455. MR 709365 (84g:65050)
 [6]
 A. S. Kronrod, Nodes and Weights for Quadrature Formulae. Sixteen Place Tables, "Nauka", Moscow, 1964; English transl., Consultants Bureau, New York, 1965. MR 0183116 (32:598)
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 C. Moler, MATLAB User's Guide, Dept. of Comput. Sci., University of New Mexico, 1981.
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 G. Monegato, "Stieltjes polynomials and related quadrature rules," SIAM Rev., v. 24, 1982, pp. 137158. MR 652464 (83d:65067)
 [9]
 T. N. L. Patterson, "The optimal addition of points to quadrature formulae," Math. Comp., v. 22, 1968, pp. 847856. MR 0242370 (39:3701)
 [10]
 R. Piessens & M. Branders, "A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type," Math. Comp., v. 28, 1974, pp. 135139. MR 0343552 (49:8293)
 [11]
 P. Turán, "On the theory of the mechanical quadrature," Acta Sci. Math. (Szeged), v. 12, 1950, pp. 3037. MR 0036797 (12:164b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407582018
PII:
S 00255718(1984)07582018
Article copyright:
© Copyright 1984 American Mathematical Society
