The Comparison of Numerical Methods for Solving Polynomial Equations

In this paper we compare the Turan process [5]-[6] with the Lehmer-Schur method [2]. We prove that the latter is better. 1. The Algorithms. We first describe the Turan process [5]-[6] which can be considered as an improvement of Graeffe's method. For the complex polynomial (1.1) P0(z)=t «/o*' = ° (fl/o G C, a00an0 * 0), 7=0 the method can be formulated as follows. Let (1.2) Pj(z) = Phx(sfz~)phx(sfi) = ¿ ak¡z' (/ = 1, 2, . . . ) fc=0 be the /th Graeffe transformation and let [ak U0/fcl_1 max — Kk 1 is fixed. Let the constants am , / be defined by the inequalities 0-5<«mo<5M°, />7T 2.5 + a arc cos 2 + 2a„ 1, m0>2. Then with the notations (1.5) M^=M\p0(z),m0], 5(°)=0, the dih step of the algorithm is the following: 1. Algorithm (T). (i) Let Sjd+i)=s(d) + 0 5(1 + amo)M(d)exvL . where / = 0, 1, . . . , / and i = V12ïïï_ + 1 Received January 14, 1976; revised August 18, 1976. AMS (MOS) subject classifications (1970). Primary 65H05. Copyright © 1978, American Mathematical Society 391 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use