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 | References | Similar Articles | Additional Information

Abstract: Orthogonal polynomials corresponding to the weight function 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.

**[1]**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****[2]**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****[3]**A. H. Stroud and Don Secrest,*Gaussian quadrature formulas*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR**0202312****[4]**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****[5]**P. Lambin and J. P. Vigneron,*Tables for the Gaussian computation of ∫₀^{∞}𝑥^{𝛼}𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 for values of 𝛼 varying continuously between -1 and +1*, Math. Comp.**33**(1979), no. 146, 805–811. MR**521295**, https://doi.org/10.1090/S0025-5718-1979-0521295-7**[6]**M. M. MORSE & H. FESHBACH,*Methods of Theoretical Physics*, McGraw-Hill, New York, 1953, Part I, Chap. 8, p. 928.**[7]**Walter Gautschi,*Construction of Gauss-Christoffel quadrature formulas*, Math. Comp.**22**(1968), 251–270. MR**0228171**, https://doi.org/10.1090/S0025-5718-1968-0228171-0**[8]**Walter Gautschi,*On the construction of Gaussian quadrature rules from modified moments.*, Math. Comp.**24**(1970), 245–260. MR**0285117**, https://doi.org/10.1090/S0025-5718-1970-0285117-6

Walter Gautschi,*Tables of Gaussian quadrature rules for the calculation of Fourier coefficients*, Math. Comp.**24**(1970), no. 110, loose microfiche suppl., A-D. MR**0285118**, https://doi.org/10.2307/2004475**[9]**W. GAUTSCHI, "Algorithms 3311 Gaussian quadrature methods,"*Comm. ACM*, v. 11, 1968, pp. 432-436.**[10]**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**, https://doi.org/10.1090/S0025-5718-69-99647-1**[11]**J. H. Wilkinson,*The evaluation of the zeros of ill-conditioned polynomials. I, II*, Numer. Math.**1**(1959), 150–180. MR**0109435**, https://doi.org/10.1007/BF01386381

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0583507-1

Article copyright:
© Copyright 1980
American Mathematical Society