NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL arXiv:1501.05033v1 [math.NA] 21 Jan 2015 I. Introduction JUN-SEOP SONG A,$_{∗}$ $^{A}$COLLEGE OF MEDICINE, YONSEI UNIVERSITY, SEOUL, REPUBLIC OF KOREA Abstract. The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the Weierstrass method to the zero in the trapezoidal Newton’s method, and prove the convergence of the method. We also present two modified Newton’s methods combined with the derivative-free method, which are constructed by applying the derivative-free method to the zero in the trapezoidal Newton’s method and the midpoint Newton’s method, respectively. Finally, we give a numerical comparison between various simultaneous methods for finding zeros of a polynomial. With a typical iteration method such as Newton’s method, an initial approximation of a zero converges to a specific zero, but the Weierstrass method (or Durand-Kerner method) approximates all simple (real or complex) zeros of polynomial simultaneously (see [2, 4]). Let P ( z ) = z n + a$_{1}$z n − 1 + · · · + a$_{n}$$_{−}$$_{1}$z + a$_{n}$ be a polynomial of degree n having simple zeros with constants a$_{1}$, a$_{2}$, . . . , a$_{n}$ . Let r$_{1}$, . . . , r$_{n}$ be the distinct zeros of P ( z ) and let distinct complex numbers z$_{1}$, . . . , z$_{n}$ be their approximations. The Weierstrass method (Durand-Kerner method) is defined as z m +1 i = z m i − P ( z m i ) ∏ j = $_{i}$( z m i − z m j ) , (1) 2010 Mathematics Subject Classification. 65H04, 65H05. Key words and phrases. Polynomial zeros, Simultaneous methods, Weierstrass method, Convergence, Newton’s method. *Corresponding author. 1