共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 127–143. 相似文献
8.
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
9.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2005,127(3):2024-2032
For polynomial matrices of full rank, including matrices of the form A - I and A - B, numerical methods for solving the following problems are suggested: find the divisors of a polynomial matrix whose spectra coincide with the zeros of known divisors of its characteristic polynomial; compute the greatest common divisor of a sequence of polynomial matrices; solve the inverse eigenvalue problem for a polynomial matrix. The methods proposed are based on the W and V factorizations of polynomial matrices. Applications of these methods to the solution of certain algebraic problems are considered. Bibliography: 3 titles._________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 122–138. 相似文献
10.
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
11.
The paper discusses the method of rank factorization for solving spectral problems for two-parameter polynomial matrices.
New forms of rank factorization, which are computed using unimodular matrices only, are suggested. Applications of these factorizations
to solving spectral problems for two-parameter polynomial matrices of both general and special forms are presented. In particular,
matrices free of the singular spectrum are considered. Conditions sufficient for a matrix to be free of the singular spectrum
and also conditions sufficient for a basis matrix of the null-space to have neither the finite regular nor the finite singular
spectrum are provided. Bibliography: 3 titles. 相似文献
12.
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.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 144–153. 相似文献
13.
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
14.
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.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 149–164. 相似文献
15.
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
16.
This paper initiates the work pertaining to the preparation of a library of MATLAB functions. This library involves programs
for personal computers, which realize methods and algorithms for solving spectral problems for polynomial one-parameter matrices,
as well as some spectral problems for two-parameter polynomial matrices. Bibliography:4 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 53–80.
Translated by N. B. Lebedinskaya. 相似文献
17.
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.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 182–214. 相似文献
18.
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.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173. 相似文献
19.
Properties of the method of ΔW-q factorization of multiparameter polynomial matrices are analyzed. Modifications of the method,
used in solving spectral and other multiparameter problems of algebra, are discussed. Bibliogrhaphy: 11 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 154–165. 相似文献
20.
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