A note on extended Gaussian quadrature rules
Author:
Giovanni Monegato
Journal:
Math. Comp. 30 (1976), 812817
MSC:
Primary 65D30
MathSciNet review:
0440878
Fulltext 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 GaussChebyshev
quadrature formula of the closed type, Math.
Comp. 22 (1968),
889–891. MR 0239756
(39 #1113), http://dx.doi.org/10.1090/S0025571819680239756X
 [2]
Philip
J. Davis and Philip
Rabinowitz, Methods of numerical integration, Academic Press
[A subsidiary of Harcourt Brace Jovanovich, Publishers] New YorkLondon,
1975. Computer Science and Applied Mathematics. MR 0448814
(56 #7119)
 [3]
Aleksandr
Semenovich Kronrod, Nodes and weights of quadrature formulas.
Sixteenplace tables, Authorized translation from the Russian,
Consultants Bureau, New York, 1965. MR 0183116
(32 #598)
 [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 (39 #3701), http://dx.doi.org/10.1090/S0025571868998669
 [5]
T. N. L. PATTERSON, "Algorithm 468Algorithm for automatic numerical integration over a finite interval," Comm. ACM, v. 16, 1973, pp. 694699.
 [6]
R. PIESSENS, "An algorithm for automatic integration," Angewandte Informatik, v. 9, 1973, pp. 399401.
 [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
(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.
110 (1935), no. 1, 501–513 (German). MR
1512952, http://dx.doi.org/10.1007/BF01448041
 [1]
 M. M. CHAWLA, "Error bounds for the GaussChebyshev quadrature formula of the closed type," Math. Comp., v. 22, 1968, pp. 889891. 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. SixteenPlace 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. 847856; Addendum, ibid., v. 22, 1968, no. 104, loose microfiche suppl. C1C11. MR 39 #3701. MR 0242370 (39:3701)
 [5]
 T. N. L. PATTERSON, "Algorithm 468Algorithm for automatic numerical integration over a finite interval," Comm. ACM, v. 16, 1973, pp. 694699.
 [6]
 R. PIESSENS, "An algorithm for automatic integration," Angewandte Informatik, v. 9, 1973, pp. 399401.
 [7]
 Ju. S. RAMSKIĬ, "The improvement of a certain quadrature formula of Gauss type," Vyčisl. Prikl. Mat. (Kiev), Vyp. 22, 1974, pp. 143146. (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. 501513. MR 1512952
Similar Articles
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/S00255718197604408783
PII:
S 00255718(1976)04408783
Article copyright:
© Copyright 1976 American Mathematical Society
