 
Summary: SIAM J. NUMER. ANAL.
Vol. 1I. No. 2, April 1974
ON THE CONVERGENCE SPEED OF SOME ALGORITHMS FOR
THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL
ROOTS*
G. ALEFELD AND J. HERZBERGERt
Abstract. This note givesan analysis ofthe order ofconvergence ofsome modified Newton methods.
The modifications we are concerned with are wellknown methodsa totalstep method and a single
step methodfor refining all roots of an nthdegree polynomial simultaneously. It is shown that for
the singlestep method the Rorder of convergence, used by Ortega and Rheinboldt in [6], is at least
2 + an > 3, where an > 1 is the unique positive root of the polynomial Pn(a)= an  a  2.
1. Preliminaries. Suppose fex) is a polynomial of nth degree given by
f( )
n n 1
X = anx + an  1X +. .. + a1 X + aü,
We assurne that the coefficients aü, a1, . . . , an are cornplex numbers and that all
the roots r1, r2' . . . , rnare distinct. Let
x(Ü) x(Ü) ... x(Ü)1 , 2' , n
be approximations for the roots of f(x). To determine the roots of f(x) we consider
the following rnethods.
