A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix |
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Authors: | Joab R. Winkler Madina Hasan |
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Affiliation: | Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, United Kingdom |
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Abstract: | A non-linear structure preserving matrix method for the computation of a structured low rank approximation 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 , and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of , which leads to a linear structure preserving matrix method, or they can be incremented during the computation of , 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 may be a good structured low rank approximation of S(f,g), but 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. |
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Keywords: | Sylvester matrix Structured low rank approximation |
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