Gaussian quadrature of integrands involving the error function

Vigneron, J. P.; Lambin, Ph.

doi:10.1090/S0025-5718-1980-0583507-1

Gaussian quadrature of integrands involving the error function

Authors: J. P. Vigneron and Ph. Lambin
Journal: Math. Comp. 35 (1980), 1299-1307
MSC: Primary 65A05; Secondary 65D30
DOI: https://doi.org/10.1090/S0025-5718-1980-0583507-1
MathSciNet review: 583507
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Abstract: Orthogonal polynomials corresponding to the weight function $1 - {\operatorname {erf}}(x)$ and defined on the positive real axis are constructed. Abscissas and weight factors for the associated Gaussian quadrature are then deduced (up to 12-point formulas). The stability of the algorithm used for this particular computation is discussed. An example is provided to test the efficiency of the new Gaussian rule.

J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The theory of splines and their applications, Academic Press, New York-London, 1967. MR 0239327
George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler, Computer methods for mathematical computations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. Prentice-Hall Series in Automatic Computation. MR 0458783
A. H. Stroud and Don Secrest, Gaussian quadrature formulas, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0202312
Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
P. Lambin and J. P. Vigneron, Tables for the Gaussian computation of $\int _{0}^{\infty }\ x^{\alpha }e^{-x}f(x)dx$ for values of $\alpha $ varying continuously between $-1$ and $+1$, Math. Comp. 33 (1979), no. 146, 805–811. MR 521295, DOI https://doi.org/10.1090/S0025-5718-1979-0521295-7

M. M. MORSE & H. FESHBACH, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, Part I, Chap. 8, p. 928.

Walter Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968), 251–270. MR 228171, DOI https://doi.org/10.1090/S0025-5718-1968-0228171-0
Walter Gautschi, On the construction of Gaussian quadrature rules from modified moments, Math. Comp. 24 (1970), 245–260. MR 285117, DOI https://doi.org/10.1090/S0025-5718-1970-0285117-6

W. GAUTSCHI, "Algorithms 3311 Gaussian quadrature methods," Comm. ACM, v. 11, 1968, pp. 432-436.

Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1–A10. MR 0245201, DOI https://doi.org/10.1090/S0025-5718-69-99647-1
J. H. Wilkinson, The evaluation of the zeros of ill-conditioned polynomials. I, II, Numer. Math. 1 (1959), 150–180. MR 109435, DOI https://doi.org/10.1007/BF01386381