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, in particular, the relatively irreducible factorization and $\Delta W$-2 factorization, which are used in solving spectral problems for two-parameter polynomial matrices $F(\lambda,\mu)$. 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 $F(\lambda,\mu)$. 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 $F(\lambda,\mu)$ 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 $\Delta W$-2 method are studied. Bibl. – 4 titles.
@article{ZNSL_2008_359_a10,
author = {V. N. Kublanovskaya},
title = {To solving problems of algebra for two-parameter polynomial matrices.~1},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {107--149},
year = {2008},
volume = {359},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a10/}
}
TY - JOUR
AU - V. N. Kublanovskaya
TI - To solving problems of algebra for two-parameter polynomial matrices. 1
JO - Zapiski Nauchnykh Seminarov POMI
PY - 2008
SP - 107
EP - 149
VL - 359
UR - http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a10/
LA - ru
ID - ZNSL_2008_359_a10
ER -
%0 Journal Article
%A V. N. Kublanovskaya
%T To solving problems of algebra for two-parameter polynomial matrices. 1
%J Zapiski Nauchnykh Seminarov POMI
%D 2008
%P 107-149
%V 359
%U http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a10/
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%F ZNSL_2008_359_a10
V. N. Kublanovskaya. To solving problems of algebra for two-parameter polynomial matrices. 1. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 107-149. http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a10/
[1] V. N. Kublanovskaya, V. B. Khazanov, Chislennye metody resheniya parametricheskikh zadach algebry. Chast 1. Odnoparametricheskie zadachi, Nauka, SPb., 2004
[2] V. N. Kublanovskaya, “Solving eigenvalue problems for two-parameter polynomial matrices. Methods and algorithms”, Sov. J. Numer. Anal. Math. Modelling, 9:2 (1994), 111–120 | DOI | MR | Zbl
