Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Chebyshev approximation of $ (1+2x)\,{\rm exp}(x\sp{2})\,{\rm erfc}\,x$ in $ 0\leq x<\infty $


Authors: M. M. Shepherd and J. G. Laframboise
Journal: Math. Comp. 36 (1981), 249-253
MSC: Primary 65D20
DOI: https://doi.org/10.1090/S0025-5718-1981-0595058-X
MathSciNet review: 595058
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We have obtained a single Chebyshev expansion of the function $ f(x) = (1 + 2x)\exp ({x^2}){\text{erfc}}\,x$ in $ 0 \leqslant x < \infty $, accurate to 22 decimal digits. The presence of the factors $ (1 + 2x)\exp ({x^2})$ causes $ f(x)$ to be of order unity throughout this range, ensuring that the use of $ f(x)$ for approximating erfc x will give uniform relative accuracy for all values of x


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

  • [1] C. Hastings, Jr., Approximations for Digital Computers, Princeton Univ. Press, Princeton, N. J., 1955, p. 169. MR 0068915 (16:963e)
  • [2] C. W. Clenshaw, "Chebyshev series for mathematical functions," in National Physical Laboratory Mathematical Tables, Vol. 5, Her Majesty's Stationery Office, London, 1962. MR 0142793 (26:362)
  • [3] Y. L. Luke, The Special Functions and their Approximations (2 Volumes; see especially Vol. 2, pp. 323-324), Academic Press, New York, 1969.
  • [4] Y. L. Luke, Mathematical Functions and their Approximations, Academic Press, New York, 1975, pp. 123-124. MR 0501762 (58:19039)
  • [5] J. L. Schonfelder, "Chebyshev expansions for the error and related functions," Math. Comp., v. 32, 1978, pp. 1232-1240. MR 0494846 (58:13630)
  • [6] K. B. Oldham, "Approximations for the $ x \exp {x^2} \operatorname{erfc} x$ function," Math. Comp., v. 22, 1968, p. 454.
  • [7] G. Dahlquist & Å. Björck, Numerical Methods, Prentice-Hall, Englewood Cliffs, N. J., 1974, pp. 104-108. MR 0368379 (51:4620)
  • [8] S.-Å. Gustavson, Private communication, 1979.
  • [9] O. Perron, Die Lehre von den Kettenbrüchen, Band II, Teubner, Berlin, 1929, p. 103.
  • [10] O. Perron, Die Lehre von den Kettenbrüchen, Band I, Teubner, Berlin, 1929, p. 4.
  • [11] R. L. Burden, J. D. Faires & A. C. Reynolds, Numerical Analysis, Prindle, Weber & Schmidt, Boston, 1978, pp. 56-57. MR 0519124 (58:24827)
  • [12] L. Fox & I. B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford Univ. Press, London and New York, 1968. MR 0228149 (37:3733)
  • [13] J. L. Schonfelder, "Generation of high precision Chebyshev expansions." (In preparation.)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D20

Retrieve articles in all journals with MSC: 65D20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0595058-X
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society