Normal factorisation of polynomials and computational issues

This paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation. This procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials. In this paper, the general aspects of the algorithm are discussed and a symbolic implementation is given. The theoretical algorithm provides the basis for a numerical procedure, the general aspects of which are also discussed here. The advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. For such polynomials, robust numerical techniques based on finding roots may then be used, if it is desired to work out the usual irreducible factorisation. A detailed description of the implementation of the algorithm is presented. The problem considered here is an integral part of computations for algebraic control problems.