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**(1970), 252-261. MR**0258271 (41:2918)****[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 (33:880)****[4]**-,*Proof of global convergence of an iterative method for calculating complex zeros of a polynomial*, Notices Amer. Math. Soc.**13**(1966), 117.**[5]**-,*The calculation of zeros of polynomials and analytic functions*, Mathematical Aspects of Computer Science, Proc. Sympos. Appl. Math., vol. 19, Amer. Math. Soc., Providence, R.I., 1967, pp. 138-152. Also available as Rep. CS 36, Computer Science Department, Stanford Univ., Stanford, Calif., 1966. MR**0233965 (38:2286)****[6]**M. A. Jenkins and J. F. Traub,*A three stage algorithm for real polynomials using quadratic iteration*, Siam J. Numer. Anal.**7**(1970). MR**0279995 (43:5716)**

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

Article copyright:
© Copyright 1986
American Mathematical Society