The location of the zeros of the higher order derivatives of a polynomial
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- by Piotr Pawlowski PDF
- Proc. Amer. Math. Soc. 127 (1999), 1493-1497 Request permission
Abstract:
Let $p(z)$ be a complex polynomial of degree $n$ having $k$ zeros in a disk $D$. We deal with the problem of finding the smallest concentric disk containing $k-l$ zeros of $p^{(l)}(z)$. We obtain some estimates on the radius of this disk in general as well as in the special case, where $k$ zeros in $D$ are isolated from the other zeros of $p(z)$. We indicate an application to the root-finding algorithms.References
- Don Coppersmith and C. Andrew Neff, Roots of a polynomial and its derivatives, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994) ACM, New York, 1994, pp. 271–279. MR 1285172
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- Victor Y. Pan, New techniques for approximating complex polynomial zeros, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994) ACM, New York, 1994, pp. 260–270. MR 1285171
- V. Pan, Sequential and parallel complexity of approximate evaluation of polynomial zeros, Comput. Math. Appl. 14 (1987), no. 8, 591–622. MR 920483, DOI 10.1016/0898-1221(87)90186-6
- James Renegar, On the worst-case arithmetic complexity of approximating zeros of polynomials, J. Complexity 3 (1987), no. 2, 90–113. MR 907192, DOI 10.1016/0885-064X(87)90022-7
- Arnold Schönhage, Equation solving in terms of computational complexity, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 131–153. MR 934220
- Steve Smale, Newton’s method estimates from data at one point, The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985) Springer, New York, 1986, pp. 185–196. MR 870648
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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1476386