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1.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》1997,86(4):2866-2879
This paper is a logical continuation of the author's discussion about the solution of spectral problems for two-parameter
polynomial matrices of general type. Various rank factorization algorithms are suggested, among them the so-called minimal
factorization of a singular two-parameter polynomial matrix of degenerate rank into a product of some matrices whose ranks
are equal to the rank of the original matrix. Spectral properties of these matrices are studied. The notion of minimal factorization
is also extended to one-parameter polynomial and constant matrices. Bibliography: 13 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 94–116
Translated by V. N. Kublanovskaya. 相似文献
2.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》1999,96(3):3085-3287
In the present paper, methods and algorithms for numerical solution of spectral problems and some problems in algebra related to them for one- and two-parameter polynomial and rational matrices are considered. A survey of known methods of solving spectral problems for polynomial matrices that are based on the rank factorization of constant matrices, i.e., that apply the singular value decomposition (SVD) and the normalized decomposition (the QR factorization), is given. The approach to the construction of methods that makes use of rank factorization is extended to one- and two-parameter polynomial and rational matrices. Methods and algorithms for solving some parametric problems in algebra based on ideas of rank factorization are presented. Bibliography: 326titles.Dedicated to the memory of my son AlexanderTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 238, 1997, pp. 7–328.Translated by V. N. Kublanovskaya. 相似文献
3.
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
4.
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
5.
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
6.
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
7.
Methods for computing scalar and vector spectral characteristics of a polynomial matrix are proposed. These methods are based on determining the so-called generating vectors (eigenvectors and principal vectors) by using the method of rank factorization of polynomial matrices. The possibility of extending the methods to the case of two-parameter polynomial matrices is indicated. Bibliography: 4 titles. 相似文献
8.
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
9.
《Linear algebra and its applications》2006,412(2-3):412-440
The topic of the paper is spectral factorization of rectangular and possibly non-full-rank polynomial matrices. To each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients. The associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix. The matrix dimensions of the pencil we obtain by solving an integer linear programming (ILP) minimization problem. Then by extracting a deflating subspace of the pencil we come to the required spectral factorization. We apply the algorithm to most general-case of inner–outer factorization, regardless continuous or discrete time case, and to finding the greatest common divisor of polynomial matrices. 相似文献
10.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2011,176(1):83-92
The paper discusses the method of hereditary pencils for computing points of the regular and singular spectra of a general
two-parameter polynomial matrix. The method allows one to reduce the spectral problems considered to eigenproblems for polynomial
matrices and pencils of constant matrices. Algorithms realizing the method are suggested and justified. Bibliography: 4 titles. 相似文献
11.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2006,132(2):214-223
The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications
to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving
spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank
is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent
matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain
rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials
of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible
factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial
solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 127–143. 相似文献
12.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》1997,86(4):2920-2925
Some algorithms are suggested for constructing pseudoinverse matrices and for solving systems with rectangular matrices whose
entries depend on a parameter in polynomial and rational ways. The cases of one- and two-parameter matrices are considered.
The construction of pseudoinverse matrices are realized on the basis of rank factorization algorithms. In the case of matrices
with polynomial occurrence of parameters, these algorithms are the ΔW-1 and ΔW-2 algorithms for one- and two-parameter matrices,
respectively. In the case of matrices with rational occurrence of parameters, these algorithms are the irreducible factorization
algorithms. This paper is a continuation of the author's studies of the solution of systems with one-parameter matrices and
an extension of the results to the case of two-parameter matrices with polynomial and rational entries. Bibliography: 12 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 176–185.
This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919).
Translated by V. N. Kublanovskaya. 相似文献
13.
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
14.
This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued
in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered.
Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers)
are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial
matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible
realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography:
15 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 117–156.
This work was carried out during our visit to Sweden under the financial support of the Chalmer University of Technology in
Góterborg and the Institute of Information Processing of the University of Umeă.
Translated by V. N. Kublanovskaya. 相似文献
15.
The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter
polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible
and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography:
6 titles. 相似文献
16.
Some statements of inverse eigenvalue problems for one-parameter and multiparameter regular polynomial matrices with linear
and nonlinear dependences on spectral parameters are considered. Methods for solving inverse eigenvalue problems based on
rank factorization, exhaustion, and reduction to nonlinear equations are proposed. Bibliography: 12 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 174–192. 相似文献
17.
An approach to solving nonlinear algebraic systems. 2 总被引:1,自引:0,他引:1
New methods of solving nonlinear algebraic systems in two variables are suggested, which make it possible to find all zero-dimensional
roots without knowing initial approximations. The first method reduces the solution of nonlinear algebraic systems to eigenvalue
problems for a polynomial matrix pencil. The second method is based on the rank factorization of a two-parameter polynomial
matrix, allowing, us to compute the GCD of a set of polynomials and all zero-dimensional roots of the GCD. Bibliography: 10
titles.
Translated by V. N. Kublanovskaya
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 71–96 相似文献
18.
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
19.
V. B. Khazanov 《Journal of Mathematical Sciences》1998,89(6):1775-1800
Spectral problems for multiparameter polynomial matrices are considered. The notions of the spectrum (including those of its
finite, infinite, regular, and singular parts), of the analytic multiplicity of a point of the spectrum, of bases of null-spaces,
of Jordan s-semilattices of vectors and of generating vectors, and of the geometric and complete geometric multiplicities
of a point of the spectrum are introduced. The properties of the above characteristics are described. A method for linearizing
a polynomial matrix (with respect to one or several parameters) by passing to the accompanying pencils is suggested. The interrelations
between spectral characteristics of a polynomial matrix and those of the accompanying pencils are established. Bibliography:
12 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 284–321.
Translated by V. B. Khazanov. 相似文献
20.
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