The nearest complex polynomial with a zero in a given complex domain

Given a univariate complex polynomial f and a closed complex domain D , whose boundary C is a curve parameterized by a piecewise rational function, we propose two computational algorithms for finding a univariate complex polynomial f ̃ such that f ̃ has a zero in D and the distance between f and f ̃...

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Bibliographic Details
Published in:Theoretical computer science Vol. 412; no. 50; pp. 7029 - 7043
Main Authors: Luo, Zhongxuan, Hu, Wenyu, Sheen, Dongwoo
Format: Journal Article
Language:English
Published: Oxford Elsevier B.V 25.11.2011
Elsevier
Subjects:
[formula omitted]-norm
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Boundaries
Complex
Computation
Computer science; control theory; systems
Exact sciences and technology
Functions of a complex variable
Mathematical analysis
Mathematical models
Mathematics
Minima
Miscellaneous
Nearest polynomial
Perturbation
Rational functions
Real functions
Sciences and techniques of general use
Searching
Theoretical computing
Zero
Nearest polynomial
Zero
Perturbation
l ∞ -norm
Complex
Polynomial
Computer theory
l
Algorithm
Rational function
norm
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:Given a univariate complex polynomial f and a closed complex domain D , whose boundary C is a curve parameterized by a piecewise rational function, we propose two computational algorithms for finding a univariate complex polynomial f ̃ such that f ̃ has a zero in D and the distance between f and f ̃ is minimal. Our approach is composed of two steps. First, in the case of D consisting of one point α , we give explicit formulas of f ̃ and the minimal distance in terms of α . Next, the case of a general closed domain D is considered by using the property that a nearest polynomial f ̃ has a zero on the boundary C . The curve C is parameterized piecewisely, and on each piece we search for the minimum of the distance between f and f ̃ . At this step we exploit the explicit formula of the minimal distance as a function of a point α . Then the global minimum and the nearest polynomial are obtained by comparing the piecewise minima. Some examples are presented: one of them confirms that the distance between a nearest complex polynomial and a given polynomial is less than that between a nearest real polynomial and the given polynomial.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2011.09.016