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1.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2006,132(2):224-228
The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row
rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original
matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix,
whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm
of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters
are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining
a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors
of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 144–153. 相似文献
2.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》1998,89(6):1715-1749
An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices
polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum,
the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the
null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar
and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of
polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems
of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices.
This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for
solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12
titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246.
Translated by V. N. Kublanovskaya. 相似文献
3.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2007,141(6):1654-1662
A new method (the RP-q method) for factorizing scalar polynomials in q variables and q-parameter polynomial matrices (q ≥
1) of full rank is suggested. Applications of the algorithm to solving systems of nonlinear algebraic equations and some spectral
problems for a q-parameter polynomial matrix F (such as separation of the eigenspectrum and mixed spectrum of F, computation
of bases with prescribed spectral properties of the null-space of polynomial solutions of F, and computation of the hereditary
polynomials of F) are considered. Bibliography: 10 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 149–164. 相似文献
4.
To solving multiparameter problems of algebra. 7. The PG-q factorization method and its applications
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2006,137(3):4844-4851
The paper continues the development of rank-factorization methods for solving certain algebraic problems for multi-parameter
polynomial matrices and introduces a new rank factorization of a q-parameter polynomial m × n matrix F of full row rank (called
the PG-q factorization) of the form F = PG, where
is the greatest left divisor of F; Δ
i
(k)
i is a regular (q-k)-parameter polynomial matrix the characteristic polynomial of which is a primitive polynomial over the
ring of polynomials in q-k-1 variables, and G is a q-parameter polynomial matrix of rank m. The PG-q algorithm is suggested,
and its applications to solving some problems of algebra are presented. Bibliography: 6 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 150–163. 相似文献
5.
V. B. Khazanov 《Journal of Mathematical Sciences》2006,137(3):4862-4878
Known types of resultant matrices corresponding to one-parameter matrix polynomials are generalized to the multiparameter
case. Based on the resultant approach suggested, methods for solving the following problems for multiparameter polynomial
matrices are developed: computing a basis of the matrix range, computing a minimal basis of the right null-space, and constructing
the Jordan chains and semilattices of vectors associated with a multiple spectrum point. In solving these problems, the original
polynomial matrix is not transformed. Methods for solving other parametric problems of algebra can be developed on the basis
of the method for computing a minimal basis of the null-space of a polynomial matrix. Issues concerning the optimality of
computing the null-spaces of sparse resultant matrices and numerical precision are not considered. Bibliography: 19 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 182–214. 相似文献
6.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2010,165(5):562-573
This paper continues the series of publications devoted to surveying and developing methods for solving the following problems
for a two-parameter matrix F (λ, μ) of general form: exhausting points of the mixed regular spectrum of F (λ, μ); performing operations on polynomials in two variables (computing the GCD and LCM of a few polynomials, division of
polynomials, and factorization); computing a minimal basis of the null-space of polynomial solutions of the matrix F (λ, μ) and separation of its regular kernel; inversion and pseudo in version of polynomial and rational matrices in two variables,
and solution of systems of nonlinear algebraic equations in two unknowns. Most of the methods suggested are based on rank
factorizations of a two-parameter polynomial matrix and on the method of hereditary pencils. Bibliography: 8 titles. 相似文献
7.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2009,157(5):731-752
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.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 107–149. 相似文献
8.
The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter
polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem
of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ,
μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving
spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil
of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil
is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed.
Bibliography: 8 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207. 相似文献
9.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2009,157(5):753-760
This paper is one of the series of survey papers dedicated to the development of methods for solving problems of algebra for
two-parameter polynomial matrices of general form. The paper considers the AB-algorithm and the ∇V-2 factorization algorithm,
together with their applications. Bibliography: 4 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 150–165. 相似文献
10.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2005,127(3):2006-2015
For a q-parameter (q 2) polynomial matrix of full rank whose regular and singular spectra have no points in common, a method for computing its partial relative factorization into a product of two matrices with disjoint spectra is suggested. One of the factors is regular and is represented as a product of q matrices with disjoint spectra. The spectrum of each of the factors is independent of one of the parameters and forms in the space q a cylindrical manifold w.r.t. this parameter. The method is applied to computing zeros of the minimal polynomial with the corresponding eigenvectors. An application of the method to computing a specific basis of the null-space of polynomial solutions of the matrix is considered. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 89–107. 相似文献
11.
12.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2007,141(6):1663-1667
A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested.
Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of
the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial
in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine
the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing
a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the
characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered.
Bibliography: 5 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173. 相似文献
13.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2006,137(3):4835-4843
For a q-parameter polynomial m × n matrix F of rank ρ, solutions of the equation Fx = 0 at points of the spectrum of the matrix
F determined by the (q −1)-dimensional solutions of the system Z[F] = 0 are considered. Here, Z[F] is the polynomial vector
whose components are all possible minors of order ρ of the matrix F. A classification of spectral pairs in terms of the matrix
A[F], with which the vector Z[F] is associated, is suggested. For matrices F of full rank, a classification and properties
of spectral pairs in terms of the so-called levels of heredity of points of the spectrum of F are also presented. Bibliography:
4 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 132–149. 相似文献
14.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2011,176(1):93-101
The method of hereditary pencils, originally suggested by the author for solving spectral problems for two-parameter matrices
(pencils of matrices), is extended to the case of q-parameter, q ≥ 2, polynomial matrices. Algorithms for computing points of the finite regular and singular spectra of a q-parameter polynomial matrix and their theoretical justification are presented. Bibliography: 2 titles. 相似文献
15.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2005,127(3):2016-2023
Methods for computing polynomials (complete polynomials) whose zeros form cylindrical manifolds of the regular spectrum of a q-parameter polynomial matrix in the space q are considered. Based on the method of partial relative factorization of matrices, new methods for computing cylindrical manifolds are suggested. The W and V methods, previously proposed for computing complete polynomials of q-parameter polynomial matrices whose regular spectrum is independent of one of the parameters, are extended to a wider class of matrices. Bibliography: 4 titles._______Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 108–121. 相似文献
16.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》1998,89(6):1694-1714
An approach to constructing methods for solving systems of nonlinear algebraic equations in three variables (SNAEs-3) is suggested.
This approach is based on the interrelationship between solutions of SNAEs-3, and solutions of spectral problems for two-
and three-parameter polynomial matrices and for pencils of two-parameter matrices. Methods for computing all of the finite
zero-dimensional roots of a SNAE-3 requiring no initial approximations of them are suggested. Some information on k-dimensional
(k>0) roots of SNAEs-3 useful for a further analysis of them is obtained. Bibliography: 17 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 159–190.
Translated by V. N. Kublanovskaya 相似文献
17.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2000,101(4):3300-3314
This paper considers the solution of a system of m nonlinear equations in q>02 variables (SNAE-q). A method for finding all
of the finite zero-dimensional roots of a given SNAE-q, which extends the method suggested previously for q=2 and q=3 to the
case q≥2, is developed and theoretically justified. This method is based on the algorithm of the ΔW-q factorization of a polynomial
q-parameter matrix and on the algorithm of relative factorization of a scalar polynomial in q variables. Bibliography: 7 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 124–146.
Translated by V. N. Kublanovskaya. 相似文献
18.
Methods and algorithms for the solution of spectral problems of singular and regular pencils D(λ, μ)=A(μ)-λB(μ) of polynomial
matrices A(μ) and B(μ) are suggested (the separation of continuous and discrete spectra, the computation of points of a discrete
spectrum with the corresponding, Jordan chains, the computation of minimal indices and a minimal basis of polynomial solutions,
the computation of the determinant of a regular pencil). Bibliography: 13 titles.
Translated by V. N. Kublanovskaya
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 26–70 相似文献
19.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2004,121(4):2508-2510
An algorithm for computing the invariant polynomials and the canonical triangular (trapezoidal) matrix for a polynomial matrix of full column rank is suggested. The algorithm is based on the Δ W-1 rank-factorization method for solving algebraic problems for polynomial matrices, previously suggested by the author. Bibliography: 3 titles. 相似文献
20.
The inversion of polynomial and rational matrices is considered. For regular matrices, three algorithms for computing the
inverse matrix in a factored form are proposed. For singular matrices, algorithms of constructing pseudoinverse matrices are
considered. The algorithms of inversion of rational matrices are based on the minimal factorization which reduces the problem
to the inversion of polynomial matrices. A class of special polynomial matrices is regarded whose inverse matrices are also
polynomial matrices. Inversion algorithms are applied to the solution of systems with polynomial and rational matrices. Bibliography:
3 titles.
Translated by V. N. Kublanovskaya.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 97–109. 相似文献