Chebyshev Approximations for the ExponentialIntegral Ei(#)

By W. J. Cody* and Henry C. Thacher, Jr.**

Abstract. The computation of the exponential integral Ei(a;), x > 0, using

rational Chebyshev approximations is discussed. The necessary approximations are

presented in well-conditioned forms for the intervals (0, 6], [6, 12], [12, 24] and

[24, <x>). Maximal relative errors are as low as from 8 X 10-19 to 2 X 10~21. In addi-

tion, the value of the zero of Ei(:c) is presented to 30 decimal places. |

1. Introduction. The classical exponential integral is defined by

(1.1) Ei(x) = f e-dt= - f ~dt (x>0)J-x t J-x t

where the integral is to be interpreted as the Cauchy principal value. Except for the

sign, it represents the natural extension of the function

Ei{z) m j* ~dt= -Ei(-z) ([arg z\ < *)

to the negative real axis.

The exponential integral was studied extensively throughout the 19th century,

in the form of the integral logarithm

Y\(x) =Ei(lnz) = "f ~-dtJ o in t

which plays a significant role in number and probability theory as well as in a variety

of physical applications. Recent applications in the theories of molecular structure

and of the solid state have produced a need for methods for high-precision evaluation

of the function using automatic computers.

To meet this need, Harris [1], and Miller and Hurst [2] published tables of the

function to 18S and 16S respectively, with useful interpolation aids, for the region

between that where the Maclaurin series is convenient and that where the asymp-

totic series gives acceptable accuracy. Clenshaw [3] has published 20D tables of the

coefficients for an expansion in Chebyshev polynomials valid for x ^ 4. Unfor-

tunately, this series converges very slowly (27 terms for 20D) and gives poor relative

accuracy where Ei(z) is small. More recently, G. F. Miller has computed (but not

yet published) coefficients for Chebyshev expansions for the intervals 4 ^ x ^ 16

and 16 ^ x ^ <x>. Both of these series require 39 terms to attain 20D accuracy, again

exhibiting slow convergence.

Received August 1, 1968.* Work performed under the auspices of the U. S. Atomic Energy Commission.

** Work supported in part by the U. S. Atomic Energy Commission and, in part, by the

University of Notre Dame.

289

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290 W. J. CODY AND HENRY C. THACHER, JR.

The present work presents nearly minimax rational approximations for Ei(x),

x > 0, with accuracies up to 18S to 20S. The approximations presented here, and in

a previously published companion paper [4], thus allow efficient high-precision

computation of Ei(a;) for all real nonzero x.

2. Functional Properties. The classical Maclaurin series

(2.1) Ei(x) = 7 + lnz+ Í~/c=i KIK

(y = 0.57721 • • •, Euler's constant) is satisfactory for computation for most smallx.

It is apparent from this series that, except for the logarithmic branch point at

the origin, Ei(a;) has no singularities in the finite complex plane, but does have a

simple zero (to which an accurate approximation will be given later) in the interval

(0, 1).In the neighborhood of that zero, significance losses make (2.1) unsatisfactory

for accurate computation, and a Taylor series expansion about the zero is to be

preferred. Differentiating (1.1) and using Leibnitz' rule, we find

(2.2) Ei(z + h) = Ei(x) + — ¿ cn(x)hn (\h\ < x)x n=o

where cn(x) are the successive derivatives of ex/x up to the factor xe~x/(n + 1) !.

These are readily computable from the recurrence

, x 1 [n+1 , s 1_1

with Co = 1, which Gautschi [5] has shown to be stable unless x is large.

For large x the Maclaurin series requires an excessive number of terms and other

methods of computation are preferable. Letting

1°° e~uCo(x) = xe *Ei(:r) = T-y- du ,

•'o 1 — u/xu/x

and using the identity

i = y, (—y + ^x^n

- u/x ¿ó\x/1 — u/x ¿To \ x / 1 — u/x

we obtain

(2.3) Co Or) = Z -k + -n Cn(x)*=0 X X

where

(2.4) Cn(x) =-.{" . U"e U, du .n\ Jq 1 — u/x

This is the asymptotic series expansion for the exponential integral with an error

term. For large enough x the series without the error term is computationally useful,

particularly if economized by Chebyshev polynomials. It can be shown that for

x > n the error in neglecting the remainder in (2.3) is positive and less than n\/nn.

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CHEBYSHEV APPROXIMATIONS FOR El(x) 291

Table I

¿ im -TOO log,Q maxEi(x) - E„m(x)im

TiTxT

0 < X < 6

m ̂\

0123456

789

••••••«»«««•»««•••»««»««««a»»**««*»««***««***»***»»««*

141781

2151235

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372

93211321430

1512864C9523

215365499628753

286

994

363 444 530

9

619

12481515

17922076

6 £ X _< 12e*&»«e*«4**ett**ttftftfc#ft*«*tt*£«fitt*eft*e*ftftfttt*ft»***&tt**tt*»tttt**«*ff*fr*«*«

113 240 320 394 480 5-77 626 742 805 862284t 309 369 446 544

309 474 577 669417 455 556 656 721462 687 735 877

1010127:

1514L763

1921

12 < X < 24i«*»*««**»»«*»***«**»*****«»*******»»,»t»,*»*«,«.«****«»,*,,,»*,*,«

.ÏÏV0 123456789

t**«»*«*»»»*»««*«»****«**«»«***«*»*.*******»**»**»****«***«****

325T438564632

303454

594+624

612f

623

796

542623

ec6876

643681785880945

690 834 855 996 1015

11671273

15031715

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5

67

89

50563174C834

462644788891979

592764896

10071127

7C7868991

11321151

809958

109611511158

899

1357

978 1054 1139 1223

14251585

16551811

Nonstandard error curve

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W. J. CODY AND HENRY C. THACHER, JR.

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W. J. CODY AND HENRY C. THACHER, JR.

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CHEBYSHEV APPROXIMATIONS FOR El(o;)

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300 W. J. CODY AND HENRY C. THACHER, JR.

3. Converging Factors. The quantities Cn(x) given by (2.4) are known as con-

verging factors. Tables and extensive theoretical discussions of these quantities are

to be found in Dingle [6], [7] and in Murnaghan and Wrench [8]. Additional ex-

pressions for the Cn(x) can be obtained from the formulations of Wadsworth [9] and

Hitotumatu [10].

Although not of primary importance in our work, expansions of the converging

factors about an arbitrary x are convenient for many purposes. First we note that

the converging factors, for each k = 1, 2, - • -, obey the relation

(3.1) Cn(x) = g ^f ' + ^4^ Cn+k(x) ,j=o nix nix

which follows from (2.3). Letting t = xu/(x + h) in (2.4) we find

1 /"oo .71 — t —th/x

Cn(x + h) = (1 + h/x)n+1 ¿ fo %[ltlx dt.

Expanding the second exponential in a Maclaurin series and using (3.1), this

becomes

«,+^(1+r[^)-s§^^].Reversing the order of summation we have

(3.2) Cn(x + A) - (l + $T \e-»Cn(x) - ± ± *=*£ &±f'1 ,

in which the double series can readily be summed by an obvious extension of

Homer's algorithm. Equation (3.2) can be shown to include for n = 0 the recur-

rences given by Harris [1] and by Miller and Hurst [2].

In addition to its utility for interpolating in a preexisting table of Cn(x), (3.2)

can also be used to generate such a table from a single initial value. Repeated ap-

plication of the expansion with positive h is numerically stable and tends to damp

out errors in the initial value.

4. Generation of the Approximations. The approximation forms and correspond-

ing intervals used are

Eim(x) = ln(x/a;o) + (x — x0)Rim(x) , 0 < x ^ 6 ,

= (ex/x)Rim(l/x) , 6 ^ x ^ 12, 12 ^ x ^ 24 ,

= (ex/x)[l + (l/x)Rlm(l/x)] , 2i^x,

where

x0 = .37250 74107 81366 63446 19918 66580

is the zero of Ei(:r), and the Rim(z) are rational functions of degree I in the numerator

and m in the denominator. The combination of forms and intervals was the best

of many such combinations tried.

The approximations were computed using standard versions of the Remez

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CHEBYSHEV APPROXIMATIONS FOR El(x) 301

algorithm for rational Chebyshev approximation [11], [12] in 25S arithmetic on a

CDC 3600. All error curves were levelled to at least 3S. Function values were com-

puted as needed using a variety of methods. For small x the standard Maclaurin

series (2.1) was used except in the vicinity of xo, where a Taylor series of the form

(2.2) was used. Harris' scheme [1] was used for 5 ^ x ^ 55 and an economized

asymptotic series for x > 55. The value of xo was determined by Newton's method

applied to (2.1) in 40S arithmetic. The same arithmetic was used to generate the

Taylor series coefficients, and the values Ei(n) necessary for Harris' scheme.

The master function routines were extensively checked for accuracy by com-

paring calculations based on two different methods wherever possible. Additional

comparisons were made against tables, especially those of Murnaghan and Wrench

[8], and against 40S computations with the Maclaurin series. We believe that the

master function routines were accurate to at least 22S, except for x > 55, where

only 20.5S was achieved.

As is indicated in our tabulated results, a number of the approximations obtained

had nonstandard error curves or gave computational difficulty because of near-

degeneracy. (See [13] for a description of this phenomenon.) The situation for the

present choice of approximation intervals is not nearly as bad, however, as the

situation when the form

(ex/x)[l + (l/x)Rlm(l/x)]

was tried for the intervals [8, =°) and [16, °°). In each case a "barrier" which made

high-accuracy approximations difficult to obtain was encountered. This barrier was

signalled by almost an entire counter-diagonal of cases in the Walsh array with non-

standard error curves. Beyond this barrier, increases in the number of coefficients

gave only minor increases in accuracy. For example, the barrier for the [16, <»)

interval occurred for I + m = 4, with the relative error for Ä22 about 2.6 X 10-8 and

that for Ä66 only 1 X 10~12. On the present [24, 00) interval, the vestige of the bar-

rier may be evident in the vicinity of the counter-diagonal for I + m — 6 (see

Table I).

5. Results. Table I lists the values of

Ei(x) - Elm(x)S im = —100 logio max

Ei(.c)

where the maximum is taken over the appropriate interval, for the initial segments

of the various L«, Walsh arrays. Tables II-V present coefficients for all approxima-

tions along the main diagonals of these arrays except for Ä44 for 24 ^ x, which has

essentially the same accuracy as Ä33.

All coefficients are given to accuracies greater than that justified by the maximal

errors, but reasonable additional rounding should not greatly affect the overall

accuracies. Each approximation listed, with the coefficients just as they appear here,

was tested against the master function routines with 5000 pseudo-random argu-

ments. In all cases the maximal error agreed in magnitude and location with the

values given by the Remez algorithm.

6. Use of the Coefficients. An attempt has been made to present the various

rational functions in a well-conditioned form. Thus, the rationals used for the

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302 W. J. CODY AND HENRY C. THACHER, JR.

interval 0 < x ^ 6 were found to lose significance during evaluation due to sub-

traction of nearly equal quantities unless expressed as ratios of finite sums of shifted

Chebyshev polynomials (see [3])

Rm(x) = ¿' PjTj*(x/6) I ¿' qjTj*(x/6) .y—o ' y-o

These sums are most conveniently evaluated by noting that T*(x) = Tt(2x—1)

and then using the Clenshaw-Rice algorithm [14]. Similarly, the remaining approxi-

mations presented in this paper are found to be well-conditioned in the /-fraction

form

Rnn(l/X) = aa + -&-/ + ~^—/ +■■■+'cti + x a2 -\- x a2 + X

For use on computers with prohibitively large divide-times, the corresponding ratio

of polynomials form may be used with losses of about 2S or 3S in some cases.

The main remaining source of avoidable error in the implementation of these

approximations into computer subroutines is in the handling of x0. If it is desired to

maintain good relative accuracy, i.e. essentially machine precision, in the vicinity

of xo, the quantity (x — x0) should be computed to higher than machine precision to

preserve the low order bits of xo- This can be readily accomplished by breaking xo

into two parts, call them x\ and x%, such that, to the precision desired, xo = xi + x%

and the floating point exponent on x2 is much less than that on x\. This breakup is

most easily accomplished by examining the octal or hexadecimal representation

xo = .27656 24522 55132 77417 11446 06004 161578

= .5F5CA 54AD2 D7F0F 264C316.

Then (x — x0) is computed as (x — x0) = (x—xi) — xi. Additional precautions will

have to be made to compute \xi(x/xo) for x c^. x0. We suggest

ln(x/x0) = ln(l + (x — x0)/x0)

coupled with a special computation of ln(l + y) for small y (usually a few terms in

the Taylor series will suffice). Note that the computation of (x—xo)/xo can be car-

ried out in normal precision once (x — xô) has been determined as above.

Subroutines based on the coefficients and techniques given here and in [4] have

been written at Argonne National Laboratory for the CDC 3600 (single precision)

and the IBM System/360 (short and long precision) computers. Essentially machine

accuracy was obtained in all cases.

Acknowledgements. The authors are indebted to W. Gautschi, H. Kuki and

E. Ng for many helpful discussions of this work. In particular, the approximation

form for the interval (0, 6] grew out of a discussion with W. Gautschi.

Argonne National Laboratory

Argonne, Illinois 60439

University of Notre Dame

Department of Computer Science

Notre Dame, Indiana 465.56

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CHEBYSHEV APPROXIMATIONS FOR El(x) 303

1. F. E. Harris, "Tables of the exponential integral Ei(i)," Math. Comp., v. 11, 1957, pp.9-16. MR 19, 464.

2. J. Miller & R. P. Hurst, "Simplified calculation of the exponential integral," Math.Comp., v. 12, 1958, pp. 187-193. MR 21 #3103.

3. C. W. Clenshaw, Chebyshev Series for Mathematical Functions, National Physical Labo-ratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962. MR 26 #362.

4. W. J. Cody & H. C. Thacher, Jr., "Rational Chebyshev approximations for the expo-nential integral £,(z)," Math. Comp., v. 22, 1968, pp. 641-649.

5. W. Gautschi, "Computation of successive derivatives of f(z)/z," Math. Comp., v. 20,1966, pp. 209-214. MR 33 #3442.

6. R. B. Dingle, "Asymptotic expansions and converging factors. I. General theory andbasic converging factors," Proc. Roy. Soc London Ser. A, v. 244, 1958, pp. 456-475. MR 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 #2146.8. 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 #8085.11. 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. 242-251.13. W. J. Cody & H. C. Thacher, Jr., "Rational Chebyshev approximations for Fermi-Dirac

integrals of orders -1/2, 1/2 and 3/2," 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.

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