A note on extended Gaussian quadrature rules

Author:
Giovanni Monegato

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

MSC:
Primary 65D30

MathSciNet review:
0440878

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.**22**(1968), 889–891. MR**0239756**, 10.1090/S0025-5718-1968-0239756-X**[2]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR**0448814****[3]**Aleksandr Semenovich Kronrod,*Nodes and weights of quadrature formulas. Sixteen-place tables*, Authorized translation from the Russian, Consultants Bureau, New York, 1965. MR**0183116****[4]**T. N. L. Patterson,*The optimum addition of points to quadrature formulae*, Math. Comp. 22 (1968), 847–856; addendum, ibid.**22**(1968), no. 104, loose microfiche supp., C1–C11. MR**0242370**, 10.1090/S0025-5718-68-99866-9**[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)**22**(1974), 143–146, 173 (Russian, with English summary). MR**0353638****[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.**110**(1935), no. 1, 501–513 (German). MR**1512952**, 10.1007/BF01448041

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

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

Article copyright:
© Copyright 1976
American Mathematical Society