Root neighborhoods of a polynomial
Mathematics of Computation
Complexity of Bezout's theorem V: polynomial time
Selected papers of the workshop on Continuous algorithms and complexity
Structured singular values and stability analysis of uncertain polynomials, part 2: a missing link
Systems & Control Letters
Approximate polynomial greatest common divisors and nearest singular polynomials
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Efficient algorithms for computing the nearest polynomial with constrained roots
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
On approximate GCDs of univariate polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Efficient algorithms for computing the nearest polynomial with a real root and related problems
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
The nearest polynomial with a given zero, and similar problems
ACM SIGSAM Bulletin
Numerical Polynomial Algebra
A note on a nearest polynomial with a given root
ACM SIGSAM Bulletin
The nearest polynomial with a given zero, revisited
ACM SIGSAM Bulletin
Locating real multiple zeros of a real interval polynomial
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Root neighborhoods, generalized lemniscates, and robust stability of dynamic systems
Applicable Algebra in Engineering, Communication and Computing
The nearest polynomial with a zero in a given domain from a geometrical viewpoint
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Real Algebraic Numbers: Complexity Analysis and Experimentation
Reliable Implementation of Real Number Algorithms: Theory and Practice
The nearest polynomial with a zero in a given domain
Theoretical Computer Science
Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric
Foundations of Computational Mathematics
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
| Hi-index | 5.23 |
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 @a, we give explicit formulas of f@? and the minimal distance in terms of @a. 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 @a. 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.