A note on extended Gaussian quadrature rules

Author:
Giovanni Monegato

Journal:
Math. Comp. **30** (1976), 812-817

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1976-0440878-3

MathSciNet review:
0440878

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Abstract: Extended Gaussian quadrature rules of the type first considered by Kronrod are examined. For a general nonnegative weight function, simple formulas for the computation of the weights are given, together with a condition for the positivity of the weights associated with the new nodes. Examples of nonexistence of these rules are exhibited for the weight functions and . Finally, two examples are given of quadrature rules which can be extended repeatedly.

**[1]**M. M. CHAWLA, "Error bounds for the Gauss-Chebyshev quadrature formula of the closed type,"*Math. Comp.*, v. 22, 1968, pp. 889-891. MR**39**#1113. MR**0239756 (39:1113)****[2]**P. J. DAVIS & P. RABINOWITZ,*Methods of Numerical Integration*, Academic Press, New York, 1975. MR**0448814 (56:7119)****[3]**A. S. KRONROD,*Nodes and Weights for Quadrature Formulae. Sixteen-Place Tables*, "Nauka", Moscow, 1964; English transl., Consultants Bureau, New York, 1965. MR**32**#597, #598. MR**0183116 (32:598)****[4]**T. N. L. PATTERSON, "The optimum addition of points to quadrature formulae,"*Math. Comp.*, v. 22, 1968, pp. 847-856; Addendum,*ibid.*, v. 22, 1968, no. 104, loose microfiche suppl. C1-C11. MR**39**#3701. MR**0242370 (39:3701)****[5]**T. N. L. PATTERSON, "Algorithm 468-Algorithm for automatic numerical integration over a finite interval,"*Comm. ACM*, v. 16, 1973, pp. 694-699.**[6]**R. PIESSENS, "An algorithm for automatic integration,"*Angewandte Informatik*, v. 9, 1973, pp. 399-401.**[7]**Ju. S. RAMSKIĬ, "The improvement of a certain quadrature formula of Gauss type,"*Vyčisl. Prikl. Mat.*(*Kiev*), Vyp. 22, 1974, pp. 143-146. (Russian) MR**50**#6121. MR**0353638 (50:6121)****[8]**W. SQUIRE,*Integration for Engineers and Scientists*, American Elsevier, New York, 1970.**[9]**G. SZEGÖ, "Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören,"*Math. Ann.*, v. 110, 1934, pp. 501-513. MR**1512952**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0440878-3

Article copyright:
© Copyright 1976
American Mathematical Society