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From: ags@s.cc.purdue.edu (Dave Seaman)
Newsgroups: sci.math,comp.graphics
Subject: Re: Solution of quartic equation
Message-ID: <2592@s.cc.purdue.edu>
Date: 25 Mar 88 17:32:36 GMT
References: <1656@thorin.cs.unc.edu> <7662@agate.BERKELEY.EDU> <2381@bsu-cs.UUCP> <726@onion.cs.reading.ac.uk>
Organization: Purdue University
Lines: 22
Keywords: quartic equation; unnecessary comlex cube roots
Summary: It ain't necessarily so.

In article <726@onion.cs.reading.ac.uk>, jadwa@henry.cs.reading.ac.uk (James Anderson) writes:
> I am an absolute beginner, so let me get this clear. If I have an
> equation with several roots and I start Newton's method lower
> than the least one it will converge to the least one and if I
> start it larger than the largest it will converge to the largest?
> No matter how close the roots are? Wow!

This is true if all roots are real and if you assume infinite precision
arithmetic (a very large assumption).  The reason is that a polynomial
with all roots real is either concave upward (if positive) or concave
downward (if negative) on the intervals (-inf,rmin] and [rmax,+inf),
where rmin and rmax are the smallest and largest roots.

The claim is not necessarily true if the polynomial has complex roots.
There may be extreme values which lie outside the interval [rmin,rmax],
and choosing an initial approximation near such an extreme may cause
the first Newton iteration to lead to a point that is far away on the
other side of all the real roots.

-- 
Dave Seaman	  					
ags@j.cc.purdue.edu