Path: news.mathworks.com!newsfeed-00.mathworks.com!newsfeed2.dallas1.level3.net!news.level3.com!postnews.google.com!o3g2000hsb.googlegroups.com!not-for-mail
From:  l_combee@yahoo.no
Newsgroups: comp.soft-sys.matlab
Subject: Re: Roots of 4th order polynomial
Date: Tue, 06 Nov 2007 08:39:03 -0800
Organization: http://groups.google.com
Lines: 41
Message-ID: <1194367143.076748.94680@o3g2000hsb.googlegroups.com>
References: <fgprvk$ao9$1@fred.mathworks.com>
NNTP-Posting-Host: 134.32.202.127
Mime-Version: 1.0
Content-Type: text/plain; charset="us-ascii"
X-Trace: posting.google.com 1194367143 22926 127.0.0.1 (6 Nov 2007 16:39:03 GMT)
X-Complaints-To: groups-abuse@google.com
NNTP-Posting-Date: Tue, 6 Nov 2007 16:39:03 +0000 (UTC)
In-Reply-To: <fgprvk$ao9$1@fred.mathworks.com>
User-Agent: G2/1.0
X-HTTP-UserAgent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.0.12) Gecko/20070508 Firefox/1.5.0.12,gzip(gfe),gzip(gfe)
Complaints-To: groups-abuse@google.com
Injection-Info: o3g2000hsb.googlegroups.com; posting-host=134.32.202.127;
Xref: news.mathworks.com comp.soft-sys.matlab:436285



On Nov 6, 2:59 pm, "Markus Buehren" <mb_matlab.REM...@gmxTHIS.de>
wrote:
> Hi!
>
> There is an analytical way to compute the roots of a
> polynomial of 4th degree, found by some guy called Ferrari.
> Here are two descriptions:
>
> http://mathworld.wolfram.com/QuarticEquation.html
>
> http://www.mathe.tu-freiberg.de/~hebisch/cafe/viertergrad.pdf
> (in german)

There is a "simpler" (?) solution then the ones described above (the
final results are obviously the same), and it doesn't
require one to "eliminate" the x^3 term.

Just rewrite as follows:

x^4 + a1*x^3 + a2*x^2 + a3*x + a4 => (x^2 + b1*x + b3)*(x^2 + b2*x +
b4)

b1,2,3,4 are simple linear and intuitive expressions in a1,2,3,4 with
an additional unknown parameter r, e.g.,
b1 = a1/2 + r
b2 = a1/2  - r
b3/2 = a2 - 1/4*a1^2 + r^2 - (...)/r
b4/2 = a2 - 1/4*a1^2 + r^2 + (...)/r

where r is a non-zero root of the cubic equation
r^3 + k*r^2 + l*r^2 + m = 0
and where k,l,m are again simple expressions of the a1,2,3,4

Voila.