Proceedings of the American Mathematical Society

Author(s): Piotr Pawlowski
Journal: Proc. Amer. Math. Soc. 127 (1999), 1493-1497.
MSC (1991): Primary 30C15; Secondary 65E05
Posted: February 4, 1999
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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.

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

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