A stability test for real polynomials

Abstract

Summary.By the argument principle the number of zeros inside of the unit circle of a real polynomial p,, p,(x) = ~=o a~ x~, a~lR, a,#O, can be estimated by the variation of the argument of p,(exp(it)) if t varies from 0 to To. This variation has its maximal value ~n iff the n distinct zeros of ~0, :--~"=oa~T~ are separated by the n--1 n -1 distinct zeros of ~p,_ 1:= ~v=oa,,+~U~. Now from Sturm sequence techniques in connection with special properties of the Chebyshev polynomials there results a very simple stability test.

Polynomial evaluation and associated polynomials

“…in some algorithms for determining or locating zeros ofq n , see, for example, [6], or in stability tests [5], [7].…”

A stability test for complex polynomials

“…", we get analogously to the real case[7] for k = 2,..., η + 1 -l7Ít-7Í°_U , ν = 0,...,k -2. great part of the coefficients fí k \ ν = 0,..., 2n -k, k = 1,..., 2n, we are interested in, results from the given coefficients yi°\ ν = 0,..., 2n, by an index shift or as a difference. The remaining triangle yi k \ ν = 0,..., 2n -k, k = η + 2,..., 2n, has to be calculated from the identities of Euclidean type <f>2n-k = (<*kUi + ßkQo)<P2n-…”

“…To overcome new technical difficulties, new ideas are necessary. At all, we propose to use our already published algorithm [7] in the real case and the algorithm formulated below in the complex case.…”

“…In a recent paper [7] we proposed an alternative test for polynomials with real coefficients which is based on Sturm sequence techniques and special properties of the Chebyshev polynomials. Now we generalize this procedure for the case of complex coefficients.…”