Summary: POLYNOMIAL ZERO FINDERS BASED ON SZEG �
O POLYNOMIALS
G. S. AMMAR \Lambda , D. CALVETTI y , W. B. GRAGG z , AND L. REICHEL x
Abstract. The computation of zeros of polynomials is a classical computational problem. This
paper presents two new zero finders that are based on the observation that, after a suitable change
of variable, any polynomial can be considered a member of a family of Szeg�o polynomials. Nu�
merical experiments indicate that these methods generally give higher accuracy than computing the
eigenvalues of the companion matrix associated with the polynomial.
Key words. Szeg�o�Hessenberg matrix, companion matrix, eigenvalue problem, continuation
method, parallel computation.
1. Introduction. The computation of the zeros of a polynomial
/n (z) = z n + ff n\Gamma1 z n\Gamma1 + \Delta \Delta \Delta + ff 1 z + ff 0 ; ff j 2 C ;
(1)
is a fundamental problem in scientific computation that arises in many diverse appli�
cations. The conditioning of this problem has been investigated by Gautschi [7, 8].
Several classical methods for determining zeros of polynomials are described by Henrici
[16, Chapter 6] and Stoer and Bulirsch [25, Chapter 5]. A recent extensive bibliogra�
phy of zero finders is provided by McNamee [20]. Among the most popular numerical
methods for computing zeros of polynomials are the Jenkins�Traub algorithm [17],
and the computation of the zeros as eigenvalues of the companion matrix

Source: Ammar, Greg - Department of Mathematical Sciences, Northern Illinois University

Collections: Mathematics