 
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 Szego 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. SzegoHessenberg 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 JenkinsTraub algorithm [17],
and the computation of the zeros as eigenvalues of the companion matrix
