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

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.