From: Per Erik Manne Newsgroups: sci.math Subject: Re: query re Hurwitz theorem on zeros of polynomials Date: Mon, 07 Sep 1998 11:52:27 +0200 Barry Brent wrote: > I'd appreciate pointers to references on the criterion of Adolph > Hurwitz for polynomials all of whose zeros have nonpositive real part. > > Please reply to barryb@primenet.com > > Thank you. > > Barry Brent Hurwitz-Routh: Let a_0 r^n + a_1 r^(n-1) + ... + a_n be a polynomial of degree n with real coefficients and a_0 > 0. A sufficient and necessary condition for all zeros of the polynomial to have negative real part is that the leading principal subdeterminants in the matrix a1 a3 a5 ... 0 0 a0 a2 a4 ... 0 0 0 a1 a3 ... 0 0 ... 0 0 0 ... a(n-1) 0 0 0 0 ... a(n-2) an are all positive. Quoted from Sydsaeter: Matematisk analyse, Bind II (in norwegian). You will probably find this also in Sydsaeter and Hammond: Mathematics for economic analysis (in english). Sydsaeter does not give a proof, but refers to Obreschkoff: Verteilung und Berechnumg der Nullstellen Reeller Polynome (in german, from 1963). -- Bergen, Per Manne ============================================================================== From: mckay@cs.concordia.ca (MCKAY john) Newsgroups: sci.math Subject: Re: query re Hurwitz theorem on zeros of polynomials Date: 7 Sep 1998 10:34:27 GMT In article barryb@primenet.com (Barry Brent) writes: >I'd appreciate pointers to references on the criterion of Adolph >Hurwitz for polynomials all of whose zeros have nonpositive real part. > >Please reply to barryb@primenet.com Thank you. Barry Brent The Routh-Hurwitz condition: See a pretty note by Strelitz in Amer. Math. Monthly ?1972. He points out that the (obviously necessary) conditions on the coeffs of the polynomial,f, and also on the polynomial, f2,(which has roots which are the sums of distinct pairs of roots of f - and obtainable by symmetric fn theory from f) are also sufficient. Undoubtedly there are faster checks. John McKay -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.