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… 
Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials
TLDR
In a simulation study, it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.
Approximate zero polynomials of polynomial matrices and linear systems
  • N. Karcanias, G. Halikias
  • Mathematics, Computer Science
    IEEE Conference on Decision and Control and European Control Conference
  • 2011
TLDR
The results provide a new definition for the “approximate”, or “almost” zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomials in a projective space.
Nearest matrix polynomials with a specified elementary divisor
TLDR
It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor and the problem is equivalent to computing a generalized notion of a structured singular value.
Approximate Polynomial Common Divisor Problem Relates to Noisy Multipolynomial Reconstruction
TLDR
An improved lattice attack to reduce both space and time costs and can be directly applied to solving the noisy multipolynomial reconstruction problem in the field of error-correcting codes.
Revisiting Approximate Polynomial Common Divisor Problem and Noisy Multipolynomial Reconstruction
TLDR
This paper presents a polynomial lattice method that can be applied directly to solve the noisy multipolynomial reconstruction problem in the field of error-correcting codes.
Structured singular value of implicit systems
Implicit systems provide a general framework in which many important properties of dynamic systems can be studied. Implicit systems are especially relevant to behavioural systems theory, the analysis
The notion of almost zeros and randomness
We investigate the problem of almost zeros of polynomial matrices as used in system theory. It is related to the controllability and observability notion of systems as well as the determination of
The feedback invariant measures of distance to uncontrollability and unobservability
ABSTRACT The selection of systems of inputs and outputs forms part of the early system design that is important since it preconditions the potential for control design. Existing methodologies for

References

SHOWING 1-10 OF 45 REFERENCES
Approximate greatest common divisor of polynomials and the structured singular value
TLDR
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
TLDR
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
TLDR
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.
A matrix pencil based numerical method for the computation of the GCD of polynomials
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
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
Numerical Computation of a Polynomial GCD and Extensions
  • V. Pan
  • Computer Science, Mathematics
  • 1996
TLDR
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
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
...
1
2
3
4
5
...