Xref: utzoo comp.theory.dynamic-sys:187 sci.math:15792 sci.physics:17523
Path: utzoo!news-server.csri.toronto.edu!cs.utexas.edu!swrinde!elroy.jpl.nasa.gov!jato!vsnyder
From: vsnyder@jato.jpl.nasa.gov (Van Snyder)
Newsgroups: comp.theory.dynamic-sys,sci.math,sci.physics
Subject: Re: Newton's method (was square roots by hand or computer)
Message-ID: <1991Mar15.201532.9300@jato.jpl.nasa.gov>
Date: 15 Mar 91 20:15:32 GMT
References: <1991Feb28.220523.6184@zaphod.mps.ohio-state.edu> <1991Mar1.001103.6341@cec1.wustl.edu> <1991Mar14.113902.97@bsu-ucs.uucp> <1991Mar15.063023.6605@dsd.es.com>
Reply-To: vsnyder@jato.Jpl.Nasa.Gov (Van Snyder)
Organization: Jet Propulsion Laboratory, Pasadena, CA
Lines: 29

In article <1991Mar15.063023.6605@dsd.es.com> rthomson@dsd.es.com (Rich Thomson) writes:
>In article <1991Mar14.113902.97@bsu-ucs.uucp>
>	00lhramer@bsu-ucs.uucp (Leslie Ramer) writes:
>>I would much rather use Newton's method to find the square root or any other
>>root for that matter. [...]
>
>Well Newton's method has some interesting properties of its own (as
>I'm sure many readers of comp.theory.dynamic-sys are aware).
>
>>Now the only problem is a starting value.  A decent starting value is 1.
>
>If you play with Newton's method long enough you come to realize that,
>in general, picking a starting value isn't always obvious.  In
>general, you pick a starting value that is "close" to the desired root
>of the function.  If you have no idea where the roots of the function
>lie, then you may be in trouble.  Newton's method may fail to
>converge.  In fact, many interesting fractal images have been computed
>for simple functions like z^3 = 0, with z being a complex number.
>
> [stuff deleted]

In the pretty inserts following page 114 of James Gleick's book "Chaos" there
is a picture of the complex boundaries of the attractors in Newton's method
for the zeroes of x^4-1=0.  Beautiful graphics.

-- 
vsnyder@jato.Jpl.Nasa.Gov
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