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From: aroberts@usq.edu.au (Tony Roberts)
Subject: Re: Accelerating Iterative Algorithms
Message-ID: <aroberts-020895093103@139.86.144.66>
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Organization: University of Southern Queensland
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Date: Wed, 2 Aug 1995 19:45:42 GMT
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In article <3vkdsp$gff@net.auckland.ac.nz>, eledscb@ccu1.auckland.ac.nz
(David Biggs) wrote:

> Most of the acceleration algorithms I have found
> are based on this main form:
> 
> 	x(k+1) = x(k) + a(k)*x'(k) 

You have to be kidding.  The above is based on the same simple minded
philosophy as that of successive over relaxation which was going out of
fashion 20 years ago.

> 
> What other techniques are normally used?

Let a sequence of approximations be simply x_{k+1}=x_k+x'_k.

If these are converging algebraically, then use Richardson extrapolation.

If these are converging geometrically, then use the iterated Shanks
transformation, unless you additionally know the geometric factors of the
convergence in which case use Romberg extrapolation.

For fun you also can try Pade approximants, via Wynn's epsilon algorithm
for example, and see if it performs better.

Most of these methods are discussed in Chapter 8 of Bender & Orszag
"Advanced mathematical methods for scientists and engineers" McGraw-Hill.


> What are their typical speedup factors?

Variable, depending upon how regular the behaviour of the sequence x_k . 
but they are so powerful that they will often work well on a DIVERGENT
sequence of approximations!  (especially if it is ocillatory divergence) 
Have fun.



                                   Tony
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