Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

MÁTYÁS , F., Bounds for the zeros of Fibonacci-like polynomials

Bounds for the zeros of Fibonacci-like polynomials 17 and (3) max IAi| < ||A||, where ||A|| denotes a norm of the matrix A. In this paper we apply the norms (4) lí-A-j^ =nmaxJöj

Jj and (5) l|A||,= /D° 2 ij I M Using the so called Gershgorin's theorem we can get a better estimation for the absolute values of the roots of f(x

) = 0 and it gives the location of zeros of /(x), too. Let us consider the sets C, of complex numbers z defined by (6) Ci - {z : Iz- an

[ < r

{} , where i = 1,2,

..., n and n (7) r

i = £la

i

Jl (n > 2). ; = i So CI is the set of complex numbers 2 which are inside the circle or on the circle with midpoint an and radius r

t in the complex plane. These sets (circles) are called to be Gershgorin-circles. Using these notations we formulate the following well-known theorem. Gershgorin's theorem. Let n > 2. For every i (1 < i < n

) there exists a j (1 < j < n) such that (8) X

l G C, and so (9) {Ai,Aj,...,A

n} cCiUC

2U---UC„.