Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Stieltjes Polynomials and Related Quadrature Formulae for a Class of Weight Functions

Authors: Walter Gautschi and Sotirios E. Notaris
Journal: Math. Comp. 65 (1996), 1257-1268
MSC (1991): Primary 33C45, 65D32
MathSciNet review: 1344614
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a (nonnegative) measure $d \sigma $ with support in the interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $\ell $, satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index $2\ell -1$, have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for $d \sigma $ have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval $[a,b]$ (under an additional assumption on $d \sigma $), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.

References [Enhancements On Off] (What's this?)

  • 1. Wm. R. Allaway, ``The identification of a class of orthogonal polynomial sets'', Ph.D. Thesis, Univ. Alberta, 1972.
  • 2. F. Caliò, W. Gautschi and E. Marchetti, ``On computing Gauss-Kronrod quadrature formulae'', Math. Comp., v. 47, 1986, pp. 639--650. MR 88a:65028
  • 3. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. MR 58:1979
  • 4. W. Gautschi, ``A survey of Gauss-Christoffel quadrature formulae'', in E. B. Christoffel (P. L. Butzer and F. Fehér, eds.), Birkhäuser, Basel, 1981, pp. 72--147. MR 83g:41031
  • 5. W. Gautschi, ``Gauss-Kronrod quadrature - a survey'', in Numerical Methods and Approximation Theory III (G. V. Milovanovi\'{c}, ed.), Faculty of Electronic Engineering, Univ. Ni\v{s}, Ni\v{s}, 1988, pp. 39--66. MR 89k:41035
  • 6. W. Gautschi and S. E. Notaris, ``Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type'', J. Comput. Appl. Math., v. 25, 1989, pp. 199--224; erratum in: J. Comput. Appl. Math., v. 27, 1989, p. 429. MR 90d:65045; 90m:65055
  • 7. W. Gautschi and T. J. Rivlin, ``A family of Gauss-Kronrod quadrature formulae'', Math. Comp., v. 51, 1988, pp. 749-754. MR 89m:65029
  • 8. A. Máté, P. Nevai and W. Van Assche, ``The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators'', Rocky Mountain J. Math., v. 21, 1991, pp. 501--527. MR 92i:42015
  • 9. G. Monegato, ``A note on extended Gaussian quadrature rules'', Math. Comp., v. 30, 1976, pp. 812--817. MR 55:13746
  • 10. G. Monegato, ``Stieltjes polynomials and related quadrature rules'', SIAM Rev., v. 24, 1982, pp. 137--158. MR 83d:65067
  • 11. S. E. Notaris, ``Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type, II'', J. Comput. Appl. Math., v. 29, 1990, pp. 161--169. MR 91b:65030
  • 12. F. Peherstorfer, ``Weight functions admitting repeated positive Kronrod quadrature'', BIT, v. 30, 1990, pp. 145--151. MR 91e:65043
  • 13. G. Szegö, Orthogonal Polynomials, Colloquium Publications, v. 23, 4th ed., American Mathematical Society, Providence, R.I., 1975. MR 51:8724

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 33C45, 65D32

Retrieve articles in all journals with MSC (1991): 33C45, 65D32

Additional Information

Walter Gautschi
Affiliation: Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907-1398

Sotirios E. Notaris
Affiliation: Department of Communications and Mass Media, National and Capodistrian University of Athens, GR-10562, Athens, Greece

Keywords: Stieltjes polynomials, Gauss-Kronrod quadrature formulae, interpolatory quadrature formulae
Received by editor(s): November 15, 1994
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society