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(f̃,g̃) of the Sylvester resultant matrix S(f,g)S(f,g) of two inexact polynomials f=f(y)f=f(y) and g=g(y)g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y)f(y) and g(y)g(y) are processed prior to the computation of S(f̃,g̃), and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of S(f̃,g̃), which leads to a linear structure preserving matrix method, or they can be incremented during the computation of S(f̃,g̃), 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)S(f,g) and that the assignment of f(y)f(y) and g(y)g(y) is important because S(f̃,g̃) may be a good structured low rank approximation of S(f,g)S(f,g), but S(g̃,f̃) may be a poor structured low rank approximation of S(g,f)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)f(y) and g(y)g(y), are shown.