Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f = f ( y ) and g = g ( y ) arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S * ( f , g ) of the Sylvester resultant matrix S ( f , g ) . In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S ( f , g ) , and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S * ( f , g ) , and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented.