Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The location of the zeros of the higher order derivatives of a polynomial


Author: Piotr Pawlowski
Journal: Proc. Amer. Math. Soc. 127 (1999), 1493-1497
MSC (1991): Primary 30C15; Secondary 65E05
DOI: https://doi.org/10.1090/S0002-9939-99-04695-X
Published electronically: February 4, 1999
MathSciNet review: 1476386
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\displaystyle{p(z)}$ be a complex polynomial of degree $\displaystyle{n}$ having $\displaystyle{k}$ zeros in a disk $\displaystyle{D}$. We deal with the problem of finding the smallest concentric disk containing $\displaystyle{k-l}$ zeros of $\displaystyle{p^{(l)}(z)}$. We obtain some estimates on the radius of this disk in general as well as in the special case, where $\displaystyle{k}$ zeros in $\displaystyle{D}$ are isolated from the other zeros of $\displaystyle{p(z)}$. We indicate an application to the root-finding algorithms.


References [Enhancements On Off] (What's this?)

  • 1. D. Coppersmith, C.A. Neff, Roots of a Polynomial and its Derivatives. Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), 271-279, ACM, New York, 1994. MR 95c:30008
  • 2. M. Marden, Geometry of Polynomials. Math. Surveys 3, Amer. Math. Soc. Providence, R.I. 1966. MR 37:1562
  • 3. V.Y. Pan, New Techniques for Approximating Complex Polynomial Zeros Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), 260-270, ACM, New York, 1994. MR 95g:65194
  • 4. V.Y. Pan, Sequential and Parallel Complexity of Approximate Evaluation of Polynomial Zeros Computers and Math. (with Applications), 14 (1987) 8, 591-622. MR 88j:65101
  • 5. J. Renegar, On the worst-case arithmetic complexity of approximating zeros of polynomials. Journal of Complexity, 3 (1987) 90-113. MR 89a:68107
  • 6. A. Sch\H{o}nhage, Equation Solving in Terms of Computational Complexity, Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986. MR 89h:68071
  • 7. S. Smale, Newton's method estimates from data at one point, The Merging Disciplines: New Directions in Pure, Applied and Computational Mathematics. 185-196, Springer-Verlag, 1986. MR 88e:65076

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30C15, 65E05

Retrieve articles in all journals with MSC (1991): 30C15, 65E05


Additional Information

Piotr Pawlowski
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Address at time of publication: Summit Systems, Inc., 22 Cortlandt Street, New York, New York 10007
Email: ppawlows@mcs.kent.edu, piotr_pawlowski@summithq.com

DOI: https://doi.org/10.1090/S0002-9939-99-04695-X
Received by editor(s): February 5, 1997
Received by editor(s) in revised form: September 3, 1997
Published electronically: February 4, 1999
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society