This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, µ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, µ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, µ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles.
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V. N. Kublanovskaya, “Methods and algorithms for solving spectral problems for polynomial and rational matrices,” Zap. Nauchn. Semin. POMI, 238, 3–330 (1997).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 107–149.
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Kublanovskaya, V.N. To solving problems of algebra for two-parameter matrices. I. J Math Sci 157, 731–752 (2009). https://doi.org/10.1007/s10958-009-9357-7
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DOI: https://doi.org/10.1007/s10958-009-9357-7