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.     

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.     

On spectral properties of multiparameter polynomial matrices

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.     

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.     

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.     

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.     

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.     

Spectral problem for polynomial matrix pencils. 2

For an arbitrary polynomial pencil of matrices A_i of dimensions m×n one presents an algorithm for the computation of the eigenvalues of the regular kernel of the pencil. The algorithm allows to construct a regular pencil having the same eigenvalues as the regular kernel of the initial pencil or (in the case of a dead end termination) allows to pass from the initial pencil to a pencil of smaller dimensions whose regular kernel has the same eigenvalues as the initial pencil. The problem is solved by reducing the obtained pencil to a linear one. For solving the problem in the case of a linear pencil one considers algorithms for pencils of full column rank as well as for completely singular pencils.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 109–116, 1981.     

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     

Uniqueness of the “nonsingular core” for Hermitian and other matrix pencils

Every pencil of hermitian matrices is conjunctive with a pencil of the form L ⊕ M, where L (the "minimal-indices" part) has no elementary divisors and M (the "nonsingular core") is a nonsingular pencil. Here it is shown that the conjunctivity type ofM is determined by that of L ⊕ M. The same method of proof applies to many other types of pencils, e.g. to congruence of pencils based on (1) a pair of symmetric matrices, (2) a pair of alternating matrices, or (3) a symmetric and an alternating matrix.     

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.     

Connection between the spectral problem for linear matrix pencils and some problems of algebra

We suggest a method by which the solution of systems of nonlinear algebraic equations in one and two variables can be reduced to the spectral problem for linear pencils of two matrices and for a system of two matrix pencils of two matrices, respectively. This method is substantially different from the traditional methods of solution; at the same time it is useful for the study of the solvability and the determination of the number of solutions of such systems. We propose a modification of the well-known elimination method for the solution of nonlinear algebraic systems of two equations in two unknowns which provides a new approach to studying and solving the problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 98–116, 1978.     

On the Computation of Particular Eigenvectors of Hamiltonian Matrix Pencils

We discuss a structure-preserving algorithm for the accurate solution of generalized eigenvalue problems for skew-Hamiltonian/Hamiltonian matrix pencils λN − ℋ. By embedding the matrix pencil λ𝒩 − ℋ into a skew-Hamiltonian/Hamiltonian matrix pencil of double size it is possible to avoid the problem of non-existence of a structured Schur form. For these embedded matrix pencils we can compute a particular condensed form to accurately compute the simple, finite, purely imaginary eigenvalues of λ𝒩 − ℋ. In this paper we describe a new method to compute also the corresponding eigenvectors by using the information contained in the condensed form of the embedded matrix pencils and associated transformation matrices. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Construction of a fundamental series of solutions of a pencil of matrices

Solution of spectral problems for a singular polynomial pencil of matrices D () of degree s1 and sizem×n is considered. Two algorithms for constructing polynomials solutions of pencils D () are considered: the first is a modification of an algorithm proposed earlier by one of the authors for determining polynomial solutions of a linear pencil; the second algorithm is based on other ideas and consists of two steps. At the first step a finite sequence of auxiliary pencils is constructed for each of which a basis of polynomial solutions of degree zero is found. At the second step the basis so constructed are rearranged into polynomial solutions of the original polynomial pencil D(). Both algorithms make it possible to find solutions of the original pencil in order of increasing degrees. For constructing a fundamental series of solutions of the pencil D() two new algorithms are proposed which work independently with either of the algorithms mentioned above for constructing polynomial solutions by rearranging them into linearly independent solutions of the pencil.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 74–93, 1984.     

AB-Algorithm and its modifications for the spectral problems of linear pencils of matrices

Summary This paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular.     

Sums of characteristic values of compact polynomial operator pencils

We consider polynomial pencils whose coefficients are compact operators. Bounds for the sums of absolute values, real and imaginary parts of characteristic values are derived. Applications to differential and difference equations are discussed.     

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order , where is the machine precision, the new method computes the eigenvalues to full possible accuracy. Received April 8, 1996 / Revised version received December 20, 1996     

Extracting partial canonical structure for large scale eigenvalue problems

We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented. This revised version was published online in June 2006 with corrections to the Cover Date.