exponential integral (Ei, E1, En...) function

[stevengj]

This is a common special function that it would be nice to include. It already is supplied by MPFR, which gives us a BigFloat version.

[StefanKarpinski]

Does it make sense for types other than BigFloat or are the typical values too large?

[stevengj]

No, typical values are well within the range of ordinary floating-point precisions, and there are practical algorithms to compute it with ordinary precision. (Or to compute them... there are several related functions.)

[stevengj]

At some point, we should just have a Special.jl package, as there is a long list of useful special functions that can be computed reasonably efficiently, and it is not clear they all belong in Base.

Of course, we might also want to dump a whole bunch of C and Fortran implementations from SciPy and elsewhere into openlibm, in which case they would be exported from Base.

[stevengj]

See also the mailing list discussion on exponential integrals.

[stevengj]

Some potentially useful references:

• Vincent Pegoraro and Philipp Slusallek, On the Evaluation of the Complex-Valued Exponential Integral, Journal of Graphics, GPU, and Game Tools, 15(3), 183-198, 2011.
• C. Chiccoli, S. Lorenzutta, and G. Maino, A numerical method for generalized exponential integrals, Computers and Math. with Appl. 14, no. 4, 261–268 (1987). Handles arbitrary complex order.
• William Cody, Henry Thacher, Rational Chebyshev Approximations for the Exponential Integral E1(x), Mathematics of Computation, Volume 22, Number 103, July 1968, pages 641-649. William Cody, Henry Thacher, Chebyshev Approximations for the Exponential Integral Ei(x), Mathematics of Computation, Volume 23, Number 106, April 1969, pages 289-303. Only real arguments.
• D. E. Amos (1980a) Algorithm 556: Exponential integrals, ACM Trans. Math. Software 6 (3), pp. 420–428. D. E. Amos (1980b) Computation of exponential integrals, ACM Trans. Math. Software 6 (3), pp. 365–377. See also Remark on Algorithm 556, ibid 1983).
• C. Chiccoli, S. Lorenzutta and G. Maino (1990a) An algorithm for exponential integrals of real order, Computing 45 (3), pp. 269–276.
• C. Chiccoli, S. Lorenzutta and G. Maino (1990b) On a Tricomi series representation for the generalized exponential integral, Internat. J. Comput. Math. 31, pp. 257–262.
• W. Gautschi (1973) Algorithm 471: Exponential integrals, Comm. ACM 16 (12), pp. 761–763.
• I. A. Stegun and R. Zucker (1974) Automatic computing methods for special functions. II. The exponential integral En⁢(x), J. Res. Nat. Bur. Standards Sect. B 78B, pp. 199–216.
• D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument, ACM Trans. Math. Software 16 (2), pp. 178–182.

Of course, the usual copyright caveats apply: don't even look at the source code of any of these, just the equations. Especially for the ACM TOMS journal, the most evil journal in numerical analysis.

[stevengj]

I assigned the problem of implementing E₁ to double-precision accuracy in the right-half complex plain (real(z) ≥ 0) in problem set 3 of our course 18.S096 this January. In the solutions, I showed how to implement it using a combination of Taylor series and continued fractions, with some custom macros to do inlining, and got performance 5–6 times faster than the Fortran code used in SciPy.

This should be helpful as a starting point for more full-featured implementations.

[tkelman]

neat, but sounds best for inclusion in a SpecialFunctions.jl rather than Base at this point.

[stevengj]

Closed in favor of SpecialFunctions issue.