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From: 00lhramer@bsu-ucs.uucp (Leslie Ramer)
Newsgroups: sci.physics,sci.math,comp.theory.dynamic-sys
Subject: Re: Newton's method (was square roots by hand or
Message-ID: <1991Mar21.000238.194@bsu-ucs.uucp>
Date: 21 Mar 91 00:02:38 GMT
References: <1991Mar17.120702.9205@jarvis.csri.toronto.edu> <1991Mar19.023209.22985@jato.jpl.nasa.gov>
Lines: 30

In article <1991Mar19.023209.22985@jato.jpl.nasa.gov>, vsnyder@jato.jpl.nasa.gov (Van Snyder) writes:
> Most codes to find all roots of polynomials, e.g. Jenkins and Traub, use a
> process called "deflation", equivalent to synthetic division, to get rid of
> roots that have been already discovered.  I.e., after finding a1, divide the
> given polynomial by (x-a1).  They also use a variation of Newton's method
> called Miller's method, that finds complex roots.  To avoid these problems,
         ^^^^^^^^^^^^^^^
> at some expense in space, simply form the companion matrix and compute it's
> eigenvalues, using a routine from EISPACK.  This is easy since the companion
> matrix is already in Hessenberg form.

I hate to be a "Net-Picker"  :-), but it's not Miller's Method, it's
Muller's Method (At least according to my numerical analysis text
Burden & Faires, 1989, 4th ed. section 2.6, pp 73-84)
You might find it interesting to note that Muller's method uses the
square root function in it's iterations.  Of course, since this method
finds complex roots of polynomials, it's the complex square root and negative
values are "okay".

I'm sure there are better techniques to use to find the square root of a 
complex number than the (rCIS theta)^1/2 techniques, though.  :-) These
genearally involve cumbersome trig functions.

-- 

"No one runs so fast as he that is chased."
           ._
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