Xref: utzoo sci.math:3087 comp.graphics:2025 Path: utzoo!mnetor!uunet!husc6!tut.cis.ohio-state.edu!rutgers!gauss.rutgers.edu!math.rutgers.edu!bumby From: bumby@math.rutgers.edu (Richard Bumby) Newsgroups: sci.math,comp.graphics Subject: Re: Solution of quartic equation Message-ID: Date: 21 Mar 88 16:06:57 GMT References: <1656@thorin.cs.unc.edu> <7662@agate.BERKELEY.EDU> <2381@bsu-cs.UUCP> Reply-To: bumby@math.rutgers.edu (Richard Bumby) Organization: Rutgers Univ., New Brunswick, N.J. Lines: 37 Keywords: quartic equation; unnecessary comlex cube roots This discussion concerns the solution of quartic equations. The formula reducing the solution of qurtic equations to the extaction of roots was referred to as 'the direct method' and questions about implementing the complex cube root it requires were addressed in some of the articles. Here is a quote from article <2381@bsu-cs.UUCP> by cfchiesa@bsu-cs.UUCP (Christopher Chiesa): Perhaps I misunderstand you (heaven knows I didn't follow the Sturm chains and so on), but... I recall taking cube and higher roots of complex numbers quite easily by converting from "cartesian" (A+Bi) representation to "polar" (r cis Theta), and performing a few straightforward real-number calculations. This is possible, but it is totally insensitive to the topology of the real numbers, and hence to any numerical considerations. Similarly, the Sturm chains are easy (I thought I posted on that, but I'm not sure it got out), but are really used for theoretical purposes. In this problem, it is probably easy to separate the roots using other considerations. If there are 4 real roots, then Newton's method is sure to locate the largest if you start from a value larger than it. The smallest is found similarly. If there are only two real roots, this may be reliable for only one of them. Deflation to reduce to an equation of lower degree is probably safe if you check the roots in the original equation. Since I last wrote on this, I have had a chance to consult "Numerical Recipes" and find that it does not give very much information. On the other hand, this is really a numerical problem, not an algebraic one. There is no reason to believe that 'extraction of roots', which must be done numerically (frequently in software) is to be preferred to a direct rootfinding method which does not require complicated preprocesssing. -- --R. T. Bumby ** Math ** Rutgers ** New Brunswick ** (in one form or another for all kinds of mail)