Structured total least norm and approximate GCDs of inexact polynomials

Authors:
Joab R. Winkler;John D. Allan
Affiliations:
Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, UK;Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

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.