A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix

  • Authors:

  • Joab R. Winkler;Madina Hasan

  • Affiliations:

  • Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, United Kingdom;Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, United Kingdom

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

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Abstract

A non-linear structure preserving matrix method for the computation of a structured low rank approximation S(f@?,g@?) of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y) and g(y) are processed prior to the computation of S(f@?,g@?), and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of S(f@?,g@?), which leads to a linear structure preserving matrix method, or they can be incremented during the computation of S(f@?,g@?), which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S(f,g) and that the assignment of f(y) and g(y) is important because S(f@?,g@?) may be a good structured low rank approximation of S(f,g), but S(g@?,f@?) may be a poor structured low rank approximation of S(g,f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of f(y) and g(y), are shown.