An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions

Authors:
Walter Gautschi and Sotirios E. Notaris

Journal:
Math. Comp. **51** (1988), 231-248

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942152-3

MathSciNet review:
942152

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Abstract | References | Similar Articles | Additional Information

Abstract: We study Gauss-Kronrod quadrature formulae for the Jacobi weight function and its special case of the Gegenbauer weight function. We are interested in delineating regions in the -plane, resp. intervals in , for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained in ; (c) all weights positive; (d) only real nodes (not necessarily satisfying (a) and/or (b)). We determine the respective regions numerically for in the Gegenbauer case, and for in the Jacobi case, where *n* is the number of Gauss nodes. Algebraic criteria, in particular the vanishing of appropriate resultants and discriminants, are used to determine the boundaries of the regions identifying properties (a) and (d). The regions for properties (b) and (c) are found more directly. A number of conjectures are suggested by the numerical results. Finally, the Gauss-Kronrod formula for the weight is obtained from the one for the weight , and similarly, the Gauss-Kronrod formula with an odd number of Gauss nodes for the weight function is derived from the Gauss-Kronrod formula for the weight .

**[1]**Franca Caliò, Walter Gautschi, and Elena Marchetti,*On computing Gauss-Kronrod quadrature formulae*, Math. Comp.**47**(1986), no. 176, 639–650, S57–S63. MR**856708**, https://doi.org/10.1090/S0025-5718-1986-0856708-8**[2]**J. J. Dongarra, C. B. Moler, J. R. Bunch & G. W. Stewart,*LINPACK Users' Guide*, SIAM, Philadelphia, Pa., 1979.**[3]**Walter Gautschi,*A survey of Gauss-Christoffel quadrature formulae*, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR**661060****[4]**K. V. Laščenov,*On a class of orthogonal polynomials*, Leningrad. Gos. Ped. Inst. Uč. Zap.**89**(1953), 167–189 (Russian). MR**0075340****[5]**Giovanni Monegato,*A note on extended Gaussian quadrature rules*, Math. Comp.**30**(1976), no. 136, 812–817. MR**440878**, https://doi.org/10.1090/S0025-5718-1976-0440878-3**[6]**Giovanni Monegato,*Stieltjes polynomials and related quadrature rules*, SIAM Rev.**24**(1982), no. 2, 137–158. MR**652464**, https://doi.org/10.1137/1024039**[7]**L. N. Puolokainen,*On the Zeros of Orthogonal Polynomials in the Case of a Sign-Variable Weight Function of Special Form*, Diploma paper, Leningrad. Gos. Univ., 1964. (Russian)**[8]**Philip Rabinowitz,*Gauss-Kronrod integration rules for Cauchy principal value integrals*, Math. Comp.**41**(1983), no. 163, 63–78. MR**701624**, https://doi.org/10.1090/S0025-5718-1983-0701624-2**[9]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., v. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.**[10]**B. L. van der Waerden,*Algebra*, vol. 1, Ungar, New York, 1970.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942152-3

Keywords:
Gauss-Kronrod quadrature formulae,
orthogonal polynomials

Article copyright:
© Copyright 1988
American Mathematical Society