The computation of the greatest common divisor (GCD) of many polynomials is a nongeneric problem. Techniques defining "approximate GCD" solutions have been defined, but the proper definition of the "approximate" GCD, and the way we can measure the strength of the approximation has remained open. This paper uses recent results on the representation of the GCD of many polynomials, in terms of factorisation of generalised resultants, to define the notion of "approximate GCD" and define the strength of any given approximation by solving an optimisation problem. The newly established framework is used to evaluate the performance of alternative procedures which have been used for defining approximate GCDs. (~)

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In this paper, a new characterisation of the approximate GCD of many polynomials is given that also allows the evaluation of accuracy of the corresponding 'approximate GCD computation'. This new approach is based on some recent results on the factorisation of the generalised resultant of a set of polynomials into reduced resultants and appropriate Toeplitz matrices representing the exact GCD [1]. This allows the reduction of 'approximate GCD' computation to an equivalent 'approximate factorisation' of generalised resultants. This new approach may be formulated as a structured optimization problem (distance between structured matrices). We use this new framework to evaluate the 'accuracy' of the 'approximate GCD' of a certain degree. This evaluation is equivalent to finding the minimal perturbation on the original set of polynomials, which make the selected given degree 'approximate GCD' exact for the perturbed set. The later makes precise the meaning of approximate GCD, since it relates it to the exact notion on a perturbed set.

Pseudospectra of matrix polynomials have been systematically investigated in the last years, since they provide important insights into the sensitivity of polynomial eigenvalue problems. An accurate approximation of the pseudospectrum of a matrix polynomial P (λ) by means of the standard grid method is computationally high demanding. In this paper, we propose an improvement of the grid method, which reduces the computational cost and retains the robustness and the parallelism of the method. In particular, after giving two lower bounds for the distance from a point to the boundary of the pseudospectrum of P (λ), we present two algorithms for the estimation of the pseudospectrum, using exclusion discs. Furthermore, two illustrative examples and an application of pseudospectra on elliptic (quadratic) eigenvalue problems are given.