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Mathematics of Computation

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On efficient computation and asymptotic sharpness of Kalantari's bounds for zeros of polynomials

Author: Yi Jin
Journal: Math. Comp. 75 (2006), 1905-1912
MSC (2000): Primary 12D10, 65H05, 68Q25; Secondary 05A15, 11B37
Published electronically: June 28, 2006
MathSciNet review: 2240641
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Abstract: We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree $ n$ our algorithm computes the first $ m$ bounds in Kalantari's family in $ O(mn)$ operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari's paper.

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Additional Information

Yi Jin
Affiliation: Department of Computer Science, Rutgers University, New Brunswick, New Jersey 08901

Keywords: Bounds on polynomial roots, asymptotic sharpness, generating functions, recurrence relations
Received by editor(s): January 14, 2005
Received by editor(s) in revised form: August 3, 2005
Published electronically: June 28, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.