Nearest common root of polynomials, approximate greatest common divisor and the structured singular value
@article{Halikias2013NearestCR, title={Nearest common root of polynomials, approximate greatest common divisor and the structured singular value}, author={George D. Halikias and G. Galanis and Nicos Karcanias and Efstathios Milonidis}, journal={IMA J. Math. Control. Inf.}, year={2013}, volume={30}, pages={423-442} }
In this paper the following problem is considered: given two coprime polynomials, find the smallest perturbation in the magnitude of their coefficients such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree…
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References
SHOWING 1-10 OF 45 REFERENCES
Approximate greatest common divisor of polynomials and the structured singular value
- Mathematics2003 European Control Conference (ECC)
- 2003
The generalisation of the method to calculate the approximate greatest common divisor of polynomials is discussed and it is shown that the problem is equivalent to the calculation of the structured singular value of a matrix, which can be performed using efficient existing techniques of robust control.
GREATEST COMMON DIVISOR OF SEVERAL POLYNOMIALS
- Mathematics, Computer Science
- 2015
This paper presents different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials and describes and compares numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness.
Structured Low Rank Approximation of a Sylvester Matrix
- Computer Science, Mathematics
- 2007
This work presents iterative algorithms that compute an approximate GCD and that can certify an approximate ∈-GCD when a tolerance ∈ is given on input and demonstrates the practical performance of these algorithms on a diverse set of univariate pairs of polynomials.
Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation
- Mathematics, Computer ScienceComput. Math. Appl.
- 2006
A matrix pencil based numerical method for the computation of the GCD of polynomials
- Mathematics[1992] Proceedings of the 31st IEEE Conference on Decision and Control
- 1992
The authors present a novel numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R(s), P/sub m,d/, of maximal degree d. It is based on a procedure…
QR factoring to compute the GCD of univariate approximate polynomials
- Computer Science, MathematicsIEEE Transactions on Signal Processing
- 2004
We present a stable and practical algorithm that uses QR factors of the Sylvester matrix to compute the greatest common divisor (GCD) of univariate approximate polynomials over /spl Ropf/[x] or /spl…
Structured singular values and stability analysis of uncertain polynomials, part 1: the generalized m
- Mathematics
- 1994
Numerical Computation of a Polynomial GCD and Extensions
- Computer Science, Mathematics
- 1996
The novel definition of approximate polynomial gcds is related to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, some deficiency of the latter definitions are demonstrated, and new effective sequential and parallel (RNC and NC) algorithms for computing approximate g cds and extended ccds are proposed.
On cone-invariant linear matrix inequalities
- MathematicsIEEE Trans. Autom. Control.
- 2000
An exact solution for a special class of cone-preserving linear matrix inequalities (LMIs) is developed. By using a generalized version of the classical Perron-Frobenius theorem, the optimal value is…