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SIAM J. NUMER. ANAL. Vol. 1I. No. 2, April 1974
 

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 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 [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.

  

Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe

 

Collections: Mathematics