Chebyshev approximations for the exponential integral
Authors:
W. J. Cody and Henry C. Thacher
Journal:
Math. Comp. 23 (1969), 289303
MSC:
Primary 65.25
MathSciNet review:
0242349
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Abstract: The computation of the exponential integral , , using rational Chebyshev approximations is discussed. The necessary approximations are presented in wellconditioned 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.
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0142793 (26 #362)
 [4]
W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for the exponential integral ,'' Math. Comp., v. 22, 1968, pp. 641649.
 [5]
Walter
Gautschi, Computation of successive derivatives
of 𝑓(𝑧)/𝑧, Math.
Comp. 20 (1966),
209–214. MR 0195239
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(21 #2145)
 [7]
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B. Dingle, Asymptotic expansions and converging factors. II. Error,
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Proc. Roy. Soc. London. Ser. A 244 (1958), 476–483.
MR
0103374 (21 #2146)
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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. 327328.
 [10]
Sin
Hitotumatu, Some considerations on the bestfit polynomial
approximations. II, Comment. Math. Univ. St. Paul. 14
(1966), 71–83. MR 0199945
(33 #8085)
 [11]
W. J. Cody & J. Stoer, ``Rational Chebyshev approximations using interpolation,'' Numer. Math., v. 9, 1966, pp. 177188.
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W.
J. Cody, Handbook Series Methods of Approximation: Rational
Chebyshev approximation using linear equations, Numer. Math.
12 (1968), no. 4, 242–251. MR
1553964, http://dx.doi.org/10.1007/BF02162506
 [13]
W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for FermiDirac integrals of orders , and ,'' Math. Comp., v. 21, 1967, pp. 3040.
 [14]
John
R. Rice, On the conditioning of polynomial and rational forms,
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 [1]
 F. E. Harris, ``Tables of the exponential integral , Math. Comp., v. 11, 1957, pp. 916. 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. 187193. 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. 641649.
 [5]
 W. Gautschi, ``Computation of successive derivatives of ,'' Math. Comp., v. 20, 1966, pp. 209214. 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. 456475. 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. 476483. 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. 327328.
 [10]
 S. Hitotumatu, ``Some considerations on the bestfit polynomial approximations. II,'' Comment. Math. Univ. St. Paul, v. 14, 1966, pp. 7183. MR 33 #8085. MR 0199945 (33:8085)
 [11]
 W. J. Cody & J. Stoer, ``Rational Chebyshev approximations using interpolation,'' Numer. Math., v. 9, 1966, pp. 177188.
 [12]
 W. J. Cody, W. Fraser & J. F. Hart, ``Rational Chebyshev approximation using linear equations,'' Numer. Math., v. 12, 1968, pp. 242251. MR 1553964
 [13]
 W. J. Cody & H. C. Thacher, Jr., ``Rational Chebyshev approximations for FermiDirac integrals of orders , and ,'' Math. Comp., v. 21, 1967, pp. 3040.
 [14]
 J. R. Rice, ``On the conditioning of polynomial and rational forms,'' Numer. Math., v. 7, 1965, pp. 426435. MR 32 #6710. MR 0189283 (32:6710)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718196902423492
PII:
S 00255718(1969)02423492
Article copyright:
© Copyright 1969
American Mathematical Society
