A new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of lR[s], P m • d of maximal degree d, is presented. This method is based on a recently developed theoretical algorithm (Karcanias 1987) that uses elementary transformations and shifting operations; the present algorithm takes into account the non-generic nature of GCD and thus uses steps, which minimize the introduction of additional errors and defines the GCD in an approximate sense. For a given set Pm,d, with a basis matrix Pm, the method defines first, the most orthogonal uncorrupted base P, from the rows of P«, where r = rank (Pm) '" m. By applying successively gaussian transformations and shifting, on the basis matrix P, E R,x(d+ll, we produce each time a new basis matrix P, with z = rank (P,) < r. The method terminates when the rank of P z is approximately equal to 1; the coefficient vector of the GCD is then defined as a row of the unit rank matrix P,. The method defines the exact degree of the GCD, successfully evaluates an approximate solution and works satisfactorily with large numbers of polynomials of any fixed degree.

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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SUMMARY Let H be a real finite dimensional Hilbert space and A an invertible linear operator on it. In this paper, we are interested in obtaining estimations of Tr(A−1) and of the norm of the error when solving the equation Ax = f ∈ H. These estimates are obtained by extrapolation of the moments of A. Numerical results are given, and applications are discussed. Copyright © 2011 John Wiley & Sons, Ltd.

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