anthony@utcsstat.UUCP (Anthony Ayiomamitis) (06/06/85)
Subject: roots of quintic polynomial
I am looking for equations (and references, if possible) which can be
used to determine the roots of a fifth-order polynomial. Any help would be
greatly appreciated !!!
--
{allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony
{ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthonybs@linus.UUCP (Robert D. Silverman) (06/06/85)
> Subject: roots of quintic polynomial > > I am looking for equations (and references, if possible) which can be > used to determine the roots of a fifth-order polynomial. Any help would be > greatly appreciated !!! > -- > > {allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony > {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony Organization: The Mitre Corporation, Bedford MA In article <2206@utcsstat.UUCP> you write: >Subject: roots of quintic polynomial > > I am looking for equations (and references, if possible) which can be >used to determine the roots of a fifth-order polynomial. Any help would be >greatly appreciated !!! >-- > > {allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony > {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony I hate to be the bearer of unhappy news but no such general solution of a quintic exists. A general polynomial of 5th degree or higher (in fact any polynomial) is solvable in terms of radicals only when its Galois group is solvable. This is an extremely rare event for high order polynomials. Polynomials of degree less than or equal to 4 always have a Galois group which is of prime order and hence cyclic and therefore solvable. That is why there are general solutions for these. Basically, the only way to solve high order polynomials is either: (1) numerically (2) trigonometrically (i.e. via the substitution x --> exp(i theta)) this can lead to a solution in terms of trig functions of algebraic angles, but it is not always clear how to solve the resulting equation.
ark@alice.UUCP (Andrew Koenig) (06/07/85)
> I am looking for equations (and references, if possible) which can be > used to determine the roots of a fifth-order polynomial. Any help would be > greatly appreciated !!! The roots of a quintic polynomial can generally not be expressed in closed form at all. This is the lowest degree polynomial of which this is true.
gwyn@brl-tgr.ARPA (06/08/85)
I suppose you know a general fifth-order polynomial is not solvable using radicals.. Your best bet is probably a general polynomial root-finder. Unfortunately the best of these are computationally slow..
karsh@geowhiz.UUCP (06/09/85)
In article <3827@alice.UUCP> ark@alice.UUCP (Andrew Koenig) writes: >> I am looking for equations (and references, if possible) which can be >> used to determine the roots of a fifth-order polynomial. Any help would be >> greatly appreciated !!! > >The roots of a quintic polynomial can generally not be expressed >in closed form at all. This is the lowest degree polynomial of >which this is true. It depends what you mean by "in closed form". It has been proven that the quinitc can't be solved "in radicals". But I understand that there is a solution in terms of elliptic modular functions. -- Bruce Karsh | U. Wisc. Dept. Geology and Geophysics | 1215 W Dayton, Madison, WI 53706 | This space for rent. (608) 262-1697 | {ihnp4,seismo}!uwvax!geowhiz!karsh |
cjh@petsd.UUCP (Chris Henrich) (06/10/85)
[] Robert D. Silverman writes: > Polynomials of degree less than or equal to 4 always have a Galois > group which is of prime order and hence cyclic and therefore > solvable. That is why there are general solutions for these. I don't think this is correct. The Galois group of an equation of order n is a subgroup of the group S/sub n/ of permutations on n things. For n <= 4 this group is solvable; hence the Galois group of an equation of order <= 4 is solvable. But it needn't be cyclic. A theory exists for the solution of equations of order 5 in "closed" form where the form involves an elliptic modular function and its inverse. This function plays a role in the solution analogous to the role of the exponential (and its inverse, the logarithm) in computing n-th roots for the "closed form" solutions of quadratics, cubics, and quartics. There is a book by Felix Klein, called _Lectures_On_The_ Icosahedron_, which used to be in print (Dover paperback) in an English translation. It's beautiful mathematics, if you happen to find it beautiful, but far from being a practical algorithm. Can anyone give a modern reference for the algorithmic aspects of computing Galois groups, and using them to deduce properties of equations? Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288
mascagni@acf4.UUCP (06/10/85)
/* acf4:net.math / anthony@utcsstat.UUCP (Anthony Ayiomamitis) / 3:32 am Jun 6, 1985 */ Subject: roots of quintic polynomial > I am looking for equations (and references, if possible) which can be >used to determine the roots of a fifth-order polynomial. Any help would be >greatly appreciated !!! There are no such equations for general quintic polynomials. The formulas for general cubic and quartic can be found in most reasonable mathematical handbook. There may exist closed form roots to quintic polynomials, but only polynomials whose coefficients satisfy very restrictive conditions. If you can reduce the case to quartics by computing one root by some numerical method and factoring thats a method that is nearly closed form, but root finding past fourth degree poly- nomials is an iterative procedure by nature. Michael Mascagni CIMS