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Calculation of Gauss quadrature rules


Authors: Gene H. Golub and John H. Welsch
Journal: Math. Comp. 23 (1969), 221-230
DOI: https://doi.org/10.1090/S0025-5718-69-99647-1
MathSciNet review: 0245201
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Abstract | References | Additional Information

Abstract: Several algorithms are given and compared for computing Gauss quadrature rules. It is shown that given the three term recurrence relation for the orthogonal polynomials generated by the weight function, the quadrature rule may be generated by computing the eigenvalues and first component of the orthornormalized eigenvectors of a symmetric tridiagonal matrix. An algorithm is also presented for computing the three term recurrence relation from the moments of the weight function.


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

  • [1] P. Concus, D. Cassatt, G. Jaehnig & E. Melby, ``Tables for the evaluation of $ \int_0^\infty {{x^\beta }{e^{ - x}}f(x)dx} $ by Gauss-Laguerre quadrature,'' Math. Comp., v. 17, 1963, pp. 245-256. MR 28 #1757. MR 0158534 (28:1757)
  • [2] D. Corneil, Eigenvalues and Orthogonal Eigenvectors of Real Symmetric Matrices, Institute of Computer Science Report, Univ. of Toronto, 1965.
  • [3] P. Davis & I. Polonsky, ``Numerical interpolation, differentiation, and integration,'' in Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Ser., 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing with corrections, 1965. MR 29 #4914; MR 31 #1400. MR 0167642 (29:4914)
  • [4] P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, Waltham, Mass., 1967. MR 35 #2482. MR 0211604 (35:2482)
  • [5] J. G. F. Francis, ``The $ Q - R$ transformation: A unitary analogue to the $ L - R$ transformation. I, II,'' Comput. J., v. 4, 1961/62, pp. 265-271; 332-345. MR 23 #B3143; MR 25 #744. MR 0130111 (23:B3143)
  • [6] W. Gautschi, ``Construction of Gauss-Christoffel quadrature formulas,'' Math. Comp., v. 22, 1968, pp. 251-270. MR 0228171 (37:3755)
  • [7] G. Golub & J. Welsch, Calculation of Gauss Quadrature Rules, Technical Report No. CS 81, Stanford University, 1967.
  • [8] U. Hochstrasser, ``Orthogonal polynomials,'' in Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Ser., 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964; 3rd printing with corrections, 1965. MR 29 #4914; MR 31 #1400. MR 0167642 (29:4914)
  • [9] I. P. Mysovskih, ``On the construction of cubature formulas with the smallest number of nodes,'' Dokl. Akad. Nauk. SSSR, v. 178, 1968, pp. 1252-1254 = Soviet Math. Dokl., v. 9, 1968, pp. 277-280. MR 36 #7328. MR 0224284 (36:7328)
  • [10] H. Rutishauser, ``On a modification of the $ Q - D$ algorithm with Graeffe-type convergence,'' $ Z$. Angew. Math. Phys., v. 13, 1963, pp. 493-496. MR 0251885 (40:5111)
  • [11] A. Stroud & D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 34 #2185. MR 0202312 (34:2185)
  • [12] H. Wilf, Mathematics for the Physical Sciences, Wiley, New York, 1962.


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-69-99647-1
Article copyright: © Copyright 1969 American Mathematical Society

American Mathematical Society