dduparc on Sun, 29 Mar 1998 09:28:41 +0200

 r2.0.6 : bug under Linux ? (+questions on polylog)

Dear friend of PARI-GP,

Everything in the sequel is send in the hope
of helping! The question 3 is perhaps off topics:
you are not requested to answer! (:-)

1.The following problem is perhaps exclusively
mine (due to another silly misunderstanding?)
but I think I must report it:

Although the releases 1.39, 2.0.1, 2.0.3 were
easy to compile and use, the release 2.0.6
cannot be used by an other user than "root".

If another user on the system calls gp, he

***   unknown user ~/gp
^__

***   Error in the PARI system. End of program.

At home my system is a i486 + 20 MB ram with
Linux 2.0.33 (slackware 3.2). I can have access
to other computers for experiments (Pentium +
128 MB ram + Linux 2.0.33 (slackware 3.3)) if
necessary.

2. Here is a microscopic remark on the documentation:
In the paragraph on polylog(), flag=1, there is a
misplaced index in the centered formula:
or something like that.

3. [slightly OFF TOPICS ?]
One of my projects is to write a contribution to
MuPAD 1.4 (function dilog) since numeric evaluation of dilog
in  this program is uneffective and unreliable:
a numeric evaluation of an integral. For instance
DIGITS:=20:dilog(100.0);dilog(0.01);
and
DIGITS:=30:dilog(100.0);dilog(0.01);
gives silly results.
(Warning: dilog(x) of MuPAD is dilog(1-x) of PARI-GP).

First I would write a correct (complex) evaluation for dilog(),
then for polylog() or perhaps better for
multilog(p+1,x) = int_1^x {ln()x^p \over 1-x} dx
(why not "multilog" ?). Then perhaps a lookup in the
integration engine of MuPAD for dilog() and perhaps
(why not to be optimistic) for multilog().
Of course I should have read the source of PARI on this
topic! But it is not really litterate programming :-)

Does anibody know a reference for the formulas
of Zagier in the same paragraph (polylog)? I did not see them,
as far as I know, in the "Cohen's book". Does anibody
know a reference on the numeric approximation of
polylog() ?

I've got Lewin's "Polylogarithms and associated functions"
North Holland 1982 and I did not see a formula with
"power series expansion in log(x)" as announed in
the documentation of PARI.

I tried to read the source of PARI but i did not understand
everything. For the moment my idea is to use for "multilog()"
a change of variable ln(t)=u then, since there is
a factor
{1 \over \exp{-u}-1}
in the integral, a power series with the Bernoulli numbers in
ln(z), whose convergence is 0< | \ln(z) | < 2 \Pi.
This is quite efficient for dilog but less for polylog() since
the formula giving multilog() in terms  of polylog()s has to be
inverted.

Has anybody a smart idea?

Best regards.
---
Daniel Duparc <dduparc@club-internet.fr>
29 av. de la Commune de Paris
94400 Vitry sur Seine (France)