From: rstrand@ihug.com.au
Subject: Re: Pade Approximants
Date: Mon, 30 Aug 1999 16:48:01 GMT
Newsgroups: sci.math.num-analysis

In article <37CA8D77.B016B836@medicine.bsd.uchicago.edu>,
  Patrick Fleury <pfleury@medicine.bsd.uchicago.edu> wrote:
> I have been looking for a decent on Pade Approximants but have not been
> able to find anything so far.  I am looking for something that will tell
> me when they are better than just series expansions and the best places
> to use them.
>
> Thanks for any replies.
>
> --
> ======================================================
> Patrick J. Fleury, Ph.D.            (773)-702-0517
> Section of Nephrology               W-514A
> pfleury@medicine.bsd.uchicago.edu
>
Virtually always. I'll try to give some intuitive points:

The Polynomial expansion is a special case of the rational approximation
where the denominator is 1.  Clearly this gives you more options.  Also
note that, through long division, rational approximation forms an
infinite series and hence allows it to approximate a larger class of
functions using less terms than a series.  Something more concrete is
that rational approximations can handle discontinuities far better than
a series.  In general they approximate badly behaved functions better.

The simplest way to is to create both approximations and choose the best
one, these days doing both is a fairly easy task.  Remember if you want
to compare apples with apples you both approximations should take the
same amount of computational effort.

Regards
Rob


Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.
==============================================================================

From: dreiss@!SPAMearthlink.net (David Reiss)
Subject: Re: Pade Approximants
Date: Mon, 30 Aug 1999 14:53:22 -0400
Newsgroups: sci.math.num-analysis

In article <37CA8D77.B016B836@medicine.bsd.uchicago.edu>, Patrick Fleury
<pfleury@medicine.bsd.uchicago.edu> wrote:

> I have been looking for a decent on Pade Approximants but have not been
> able to find anything so far.  I am looking for something that will tell
> me when they are better than just series expansions and the best places
> to use them.
> 
> Thanks for any replies.
> 
> --
> ======================================================
> Patrick J. Fleury, Ph.D.            (773)-702-0517
> Section of Nephrology               W-514A
> pfleury@medicine.bsd.uchicago.edu


Here are two references. The second one is classic but is out 
of print; however, you should be able to get it at a 
reasonable technical library. The first one 
has a decent chapter on Pade approximants. One thing 
about Pade approximants is that they are hard to 
characterize with regard to when they do and do not
do a good or magical job. However, the essential
hand-waving point is that, since they are rational
function approximations to a given function, they can 
capture some of the singularity behavior of the function
that a power series cannot. Both a Taylor series and a Pade
approximant make use of the same information (function 
derivatives at particular points: note that there
are "generalized Pade approximants that make use of 
the information from serveral simultaneous points
whereas the Taylor expansion doesn't do this), 
but the Pade approximant
can often make use of this information to greater
effect.

C. M. Bender and S. A. Orszag, "Advanced 
Mathematical Methods for Scientists and Engineers", 
New York: McGraw-Hill (1978).

G. A Baker, "Essentials of Pade Approximants", 
New York: Academic Press 
(1975).


Cheers,

David


   
)--------------------------------------
)        Scientific Arts:            
) Creative Services and Consultation 
) for the Applied and Pure Sciences  
)                                    
)   http://www.scientificarts.com    
)                                    
) David Reiss                        
) Email: dreiss@!SPAMscientificarts.com
) [Remove the !SPAM to send email]   
)-----------------------------------