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ID Category Severity Reproducibility Date Submitted Last Update
0000519 [ALGLIB] Spec.functions major always 2013-04-23 17:24 2018-01-02 17:23
Reporter zaytsev View Status public  
Assigned To SergeyB
Priority normal Resolution open  
Status assigned   Product Version 3.7.0
Summary 0000519: Exponential integral Ei(x) returns 0 for x <= 0
Description The Ei(x) in ALGLIB returns 0 for all x <= 0 . This is not documented on the website:

In the source code, however, it says "Not defined for x <= 0", which, of course, explains this behavior, but doesn't really help.

Anyways, the function Ei(x) is defined on all real axis [1] and I see no reason why you would silently return 0 for x <= 0, while many other special function packages allow one to compute Ei(x) for negative arguments.

I need fast and accurate routines to compute the exponential integrals and would highly appreciate if you could address this problem.


Additional Information I'm looking specifically at the C/C++ version of ALGLIB, but it shouldn't really matter.
Programming language All
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SergeyB (administrator)
2013-04-23 22:04

According to Wiki [2], "definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and .[1] In general, a branch cut is taken on the negative real axis".

So values at negative axis are not well-defined because of branch cut.

zaytsev (reporter)
2013-04-23 23:38

Hi Sergey,

The discussion about the ambiguity that you quoted above relates to the function of the complex argument (it starts with "For complex values of the argument..."), however, we are talking about a real-valued function of a real argument here.

Just have a look at the integral, it's well-defined for x < 0, however, at 0 the divergence happens and then, in order to use the same definition for x >= 0, you have to consider it as a VP.

Right now, I've switched to Boost [3], and I'm totally delighted about the accuracy and speed (4x faster then SPECFUN, didn't measure against CEPHES, because I need negative values).

However, Boost is a couple of hundreds of megabytes of source code, which sounds a bit too much if you only want one function out of it :-)



- Issue History
Date Modified Username Field Change
2013-04-23 17:24 zaytsev New Issue
2013-04-23 17:24 zaytsev Programming language => All
2013-04-23 22:04 SergeyB Note Added: 0000062
2013-04-23 23:38 zaytsev Note Added: 0000063
2018-01-02 17:23 SergeyB Status new => assigned
2018-01-02 17:23 SergeyB Assigned To => SergeyB

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