Chebyshev approximations for the exponential integral

Authors:
W. J. Cody and Henry C. Thacher

Journal:
Math. Comp. **23** (1969), 289-303

MSC:
Primary 65.25

DOI:
https://doi.org/10.1090/S0025-5718-1969-0242349-2

MathSciNet review:
0242349

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The computation of the exponential integral , , using rational Chebyshev approximations is discussed. The necessary approximations are presented in well-conditioned forms for the intervals , , and . Maximal relative errors are as low as from . In addition, the value of the zero of is presented to 30 decimal places.

**[1]**F. E. Harris, ``Tables of the exponential integral ,*Math. Comp.*, v. 11, 1957, pp. 9-16. MR**19**, 464. MR**0088069 (19:464b)****[2]**J. Miller & R. P. Hurst, ``Simplified calculation of the exponential integral,''*Math. Comp.*, v. 12, 1958, pp. 187-193. MR**21**#3103. MR**0104348 (21:3103)****[3]**C. W. Clenshaw,*Chebyshev Series for Mathematical Functions*, National Physical Laboratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962. MR**26**#362. MR**0142793 (26:362)****[4]**W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for the exponential integral ,''*Math. Comp.*, v. 22, 1968, pp. 641-649.**[5]**W. Gautschi, ``Computation of successive derivatives of ,''*Math. Comp.*, v. 20, 1966, pp. 209-214. MR**33**#3442. MR**0195239 (33:3442)****[6]**R. B. Dingle, ``Asymptotic expansions and converging factors. I. General theory and basic converging factors,''*Proc. Roy. Soc. London Ser. A*, v. 244, 1958, pp. 456-475. MR**21**#2145. MR**0103373 (21:2145)****[7]**R. B. Dingle, ``Asymptotic expansions and converging factors. II. Error, Dawson, Fresnel, exponential, sine and cosine, and similar integrals,''*Proc. Roy. Soc. London Ser. A*, v. 244, 1958, pp. 476-483. MR**21**#2146. MR**0103374 (21:2146)****[8]**F. D. Murnaghan & J. W. Wrench, Jr.,*The Converging Factor for the Exponential Integral*, David Taylor Model Basin Applied Mathematics Laboratory Report 1535, 1963.**[9]**D. van Z. Wadsworth, ``Improved asymptotic expansion for the exponential integral with positive argument,''*Math. Comp.*, v. 19, 1965, pp. 327-328.**[10]**S. Hitotumatu, ``Some considerations on the best-fit polynomial approximations. II,''*Comment. Math. Univ. St. Paul*, v. 14, 1966, pp. 71-83. MR**33**#8085. MR**0199945 (33:8085)****[11]**W. J. Cody & J. Stoer, ``Rational Chebyshev approximations using interpolation,''*Numer. Math.*, v. 9, 1966, pp. 177-188.**[12]**W. J. Cody, W. Fraser & J. F. Hart, ``Rational Chebyshev approximation using linear equations,''*Numer. Math.*, v. 12, 1968, pp. 242-251. MR**1553964****[13]**W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for Fermi-Dirac integrals of orders , and ,''*Math. Comp.*, v. 21, 1967, pp. 30-40.**[14]**J. R. Rice, ``On the conditioning of polynomial and rational forms,''*Numer. Math.*, v. 7, 1965, pp. 426-435. MR**32**#6710. MR**0189283 (32:6710)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65.25

Retrieve articles in all journals with MSC: 65.25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1969-0242349-2

Article copyright:
© Copyright 1969
American Mathematical Society