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From: wrf@mab.ecse.rpi.edu (Wm Randolph Franklin)
Newsgroups: sci.math,comp.graphics
Subject: Re: Solution to quartic eqn?
Keywords: quartic, polynomial, roots
Message-ID: <1989Nov9.225450.4723@rpi.edu>
Date: 9 Nov 89 22:54:50 GMT
References: <19606@vax5.CIT.CORNELL.EDU>
Organization: Rensselaer Polytechnic Institute, Troy NY
Lines: 25

In <19606@vax5.CIT.CORNELL.EDU> cfwy@vax5.cit.cornell.edu (Larry Gritz) writes:
>I need an algorithm to find the roots of a quartic (polynomial of degree 4)
>equation to find the solution to a ray-torus intersection test.  Quadratic
>is easy, and I found a solution to cubic, but I can't get a reference for
>quartic.  The equation looks like this:
>            a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0 = 0
>I'd appreciate a pointer to a solution, or better, a short algorithm
>(efficient is good, too).

The  standard  explicit  solution may  be   found in handbooks   such as
Abramowitz.  One disadvantage is  that it requires using complex numbers
even  if all coefficients and all  roots are real.  This includes taking
cube roots of complex numbers.

In practice, finding  any  one root by  Newton  and then deflating to  a
cubic might be fastest.

One problem with  Newton is that  it  converges very slowly  to multiple
roots.  However the multiple roots of f are  roots of gcd(f,f'),  so you
can identify them.
-- 
						   Wm. Randolph Franklin
Internet: wrf@ecse.rpi.edu (or @cs.rpi.edu)    Bitnet: Wrfrankl@Rpitsmts
Telephone: (518) 276-6077;  Telex: 6716050 RPI TROU; Fax: (518) 276-6261
Paper: ECSE Dept., 6026 JEC, Rensselaer Polytechnic Inst, Troy NY, 12180