Gaussian quadrature of integrands involving the error function
HTML articles powered by AMS MathViewer
- by J. P. Vigneron and Ph. Lambin PDF
- Math. Comp. 35 (1980), 1299-1307 Request permission
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.References
- 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 Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. 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, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. 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 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 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 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 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 10.1007/BF01386381
Additional Information
- © Copyright 1980 American Mathematical Society
- 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