Summary: SIAM J. NUMER. ANAL.
Vol. 1I. No. 2, April 1974
ON THE CONVERGENCE SPEED OF SOME ALGORITHMS FOR
THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL
G. ALEFELD AND J. HERZBERGERt
Abstract. This note givesan analysis ofthe order ofconvergence ofsome modified Newton methods.
The modifications we are concerned with are well-known methods-a total-step method and a single-
step method-for refining all roots of an nth-degree polynomial simultaneously. It is shown that for
the single-step method the R-order of convergence, used by Ortega and Rheinboldt in , 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
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.