On Some Spectral Characteristics of Multiparameter Polynomial Matrices

Definitions of certain spectral characteristics of polynomial matrices (such as the analytical (algebraic) and geometric multiplicities of a point of the spectrum, deflating subspaces, matrix solvents, and block eigenvalues and eigenvectors) are generalized to the multiparameter case, and properties of these characteristics are analyzed. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 166–173.     

To solving spectral problems for multiparameter polynomial matrices

An approach to solving spectral problems for multiparameter polynomial matrices based on passing to accompanying pencils of matrices is described. Also reduction of spectral problems for multiparameter pencils of complex matrices to the corresponding real problems is considered. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2006, pp. 212–231.     

To Solving Multiparameter Problems of Algebra. 4. The AB-Algorithm and its Applications

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.     

To solving problems of algebra for two-parameter matrices. III

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.     

The resultant approach to computing vector characteristics of multiparameter polynomial matrices

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.     

Spectral properties of λ-matrices

One investigates the spectral problem for polynomial -matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the -matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a -matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a -matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a -matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 180–194, 1981.     

Spectral problems for pencils of polynomial matrices. Methods and algorithms. V

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     

To solving spectral problems for <Emphasis Type="Italic">q</Emphasis>-parameter polynomial matrices

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.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

To solving problems of algebra for two-parameter matrices. IV

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.     

Pencils of waveguide type and related extremal problems

The variational properties of the spectra of a class of quadratic pencils are investigated. These operator pencils are not strongly damped, which is expressed in a considerable manner in its properties. The obtained results are fundamental in the investigation of two-parameter pencils of waveguide type, which model pencils arising in the theory of regular waveguides. The considerable difficulties, arising at the investigation of pencils of waveguide type, are explained by the fact that they do not generate Rayleigh systems in the entire space, but only on certain of its nonconvex homogeneous sets. These sets occur here as the sets of the admissible vectors of the corresponding extremal problems.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 80–96, 1990.     

Methods for solving spectral problems for multiparameter matrix pencils

Methods for solving the partial eigenproblem for multiparameter regular pencils of real matrices, which allow one to improve given approximations of an eigenvector and the associated point of the spectrum (both finite and infinite) are suggested. Ways of extending the methods to complex matrices, polynomial matrices, and coupled multiparameter problems are indicated. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 139–168.     

Solution of the partial eigenvalue problem for a regular matrix pencil

Algorithms are proposed for the solution of the partial eigenvalue problem for regular polynomial matrix pencils. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 33–39, 1987.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

To Solving Multiparameter Problems of Algebra. 5. The ∇V-q Factorization Algorithm and its Applications

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.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory

N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.     

An approach to solving multiparameter algebraic problems

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.     

To solving problems of algebra for two-parameter matrices. VIII

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.     

Solution of systems of nonlinear algebraic equations in three variables. Methods and algorithms. 3

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