Newton's method and the Jenkins-Traub algorithm

Author:
R. N. Pederson

Journal:
Proc. Amer. Math. Soc. **97** (1986), 687-690

MSC:
Primary 30C15

DOI:
https://doi.org/10.1090/S0002-9939-1986-0845988-2

MathSciNet review:
845988

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Abstract: In this paper we propose to show how a multi-increment version of Newton's method can be used to obtain starting points for the Jenkins-Traub algorithm.

**[1]**M. A. Jenkins,*Three-stage variable-shift iterations for the solution of polynomial equations with a posteriori error bounds for the zeros*, Dissertation, Stanford Univ., Stanford, Calif., 1969. Available as Rep. CS 138, Computer Science Department, Stanford Univ.**[2]**M. A. Jenkins and J. F. Traub,*A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration*, Numer. Math.**14**(1969/1970), 252–263. MR**0258271**, https://doi.org/10.1007/BF02163334**[3]**J. F. Traub,*A class of globally convergent iteration functions for the solution of polynomial equations*, Math. Comp.**20**(1966), 113–138. MR**0192655**, https://doi.org/10.1090/S0025-5718-1966-0192655-2**[4]**-,*Proof of global convergence of an iterative method for calculating complex zeros of a polynomial*, Notices Amer. Math. Soc.**13**(1966), 117.**[5]**J. F. Traub,*The calculation of zeros of polynomials and analytic functions*, Proc. Sympos. Appl. Math., Vol. XIX, Amer. Math. Soc., Providence, R.I., 1967, pp. 138–152. MR**0233965****[6]**M. A. Jenkins and J. F. Traub,*A three-stage algorithm for real polynomials using quadratic iteration.*, SIAM J. Numer. Anal.**7**(1970), 545–566. MR**0279995**, https://doi.org/10.1137/0707045

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0845988-2

Article copyright:
© Copyright 1986
American Mathematical Society