From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Newsgroups: sci.math.num-analysis
Subject: Re: Pade approximant for f*(z) = f(z*)
Date: 27 Oct 1998 13:57:05 GMT

In article <36351A44.6B44B6DA@cezanne.phy.queensu.ca>,
 "Robert J. Gooding" <bob@cezanne.phy.queensu.ca> writes:
|> consider a complex function that satisifies
|> 
|> f*(z) = f(z*) 
|> 
|> in the complex plane. typically one only knows this function
|> at a finite number of points along the imaginary axis, and
|> wishes to analytically continue the function onto the real axis.
|> since the functions in question often have simple pole structures,
|> the most familiar and usually reliable way to perform this
|> continuation is with a [L,M] Pade approximant, say
|> 
|> f(x) = (p0 + p1*x + p2*x^2 + ... ) / (1 + q1*x + q2*x^2 + ...)
|> 
|> Using the above relation makes clear the necessity of having
|> the Pade coefficients (p0,p1, ..., q1,q2,...) real.
snip
the Pade approximations are computed from the power series coefficients of
f, which are real in this case, hence the outcome is real. but this is
not your question, I guess.
Since you wrote about a finite number
of function values along the imaginary axes as your only information.
you could take a numerical approximation approach 
minimize   F( eps,p0,p1,..,pn,...,qm)) =  eps  (real) 

subject to the constraints  

   -eps <= abs( f(z_i) - R_{n,m}(z_i)  ) <= eps ,  i=1,...,N

where  z_i are the given points. computing "abs" analytically and 
multplying by the denominator, you arrive at a (real) system of nonlinear
inequalities as side constraints, i.e. you have an ordinary
nonlinear programming problem. If N>>n+m ,then you will have what usually
is called an finely grained discretized continuous problem.
the code CFSQP of Andre Tits is especially strong for this kind of 
problems. there remains a problem how to exclude poles 
of this approximation from (a part of) the real axes.
this could be done by imposing additional (real) constraints (lower bounds)
for the denominator
look at 
http://plato.la.asu.edu/guide.html
how to access CFSQP and other nonlinear programming codes. 
hope this helps
peter
==============================================================================
[Note: taken from Maple's  ?numapprox,pade  help command:

- The (m,n) Pade approximation is defined to be the rational function
  p(x)/q(x) with deg(p(x)) <= m and deg(q(x)) <= n such that the Taylor (or
  Laurent) series expansion of p(x)/q(x) has maximal initial agreement with
  the series expansion of f. In normal cases, the series expansion agrees
  through the term of degree m+n.