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]**Frank E. Harris,*Tables of the exponential integral 𝐸𝑖(𝑥)*, Math. Tables Aids Comput.**11**(1957), 9–16. MR**0088069**, https://doi.org/10.1090/S0025-5718-1957-0088069-0**[2]**James Miller and R. P. Hurst,*Simplified calculation of the exponential integral*, Math. Tables Aids Comput.**12**(1958), 187–193. MR**0104348**, https://doi.org/10.1090/S0025-5718-1958-0104348-3**[3]**C. W. Clenshaw,*Chebyshev series for mathematical functions*, National Physical Laboratory Mathematical Tables, Vol. 5. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London, 1962. MR**0142793****[4]**W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for the exponential integral ,''*Math. Comp.*, v. 22, 1968, pp. 641-649.**[5]**Walter Gautschi,*Computation of successive derivatives of 𝑓(𝑧)/𝑧*, Math. Comp.**20**(1966), 209–214. MR**0195239**, https://doi.org/10.1090/S0025-5718-1966-0195239-5**[6]**R. B. Dingle,*Asymptotic expansions and converging factors. I. General theory and basic converging factors*, Proc. Roy. Soc. London. Ser. A**244**(1958), 456–475. MR**0103373**, https://doi.org/10.1098/rspa.1958.0054**[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**244**(1958), 476–483. MR**0103374**, https://doi.org/10.1098/rspa.1958.0055**[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]**Sin Hitotumatu,*Some considerations on the best-fit polynomial approximations. II*, Comment. Math. Univ. St. Paul.**14**(1966), 71–83. MR**0199945****[11]**W. J. Cody & J. Stoer, ``Rational Chebyshev approximations using interpolation,''*Numer. Math.*, v. 9, 1966, pp. 177-188.**[12]**W. J. Cody,*Handbook Series Methods of Approximation: Rational Chebyshev approximation using linear equations*, Numer. Math.**12**(1968), no. 4, 242–251. MR**1553964**, https://doi.org/10.1007/BF02162506**[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]**John R. Rice,*On the conditioning of polynomial and rational forms*, Numer. Math.**7**(1965), 426–435. MR**0189283**, https://doi.org/10.1007/BF01436257

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