Inversion of polynomial and rational matrices

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.     

Methods and algorithms of solving spectral problems for polynomial and rational matrices

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.     

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. I

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.     

On some factorizations of two-parameter polynomial matrices

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.     

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.     

On irreducible factorizations of rational matrices and their applications

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.     

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.     

Modifications of the ΔW-q Factorization Method for Multiparameter Polynomial Matrices and Their Properties

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.     

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

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.     

A library of matlab functions for polynomial matrices

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.     

An approach to solving nonlinear algebraic systems. 2

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     

Solution of systems of general-form nonilinear algebraic equations. Methods and algorithms. IV

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.     

To the Solution of Partial Eigenproblems for Polynomial Matrices

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.     

To solving multiparameter problems of algebra. 7. The PG-q factorization method and its applications

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. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 150–163.     

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

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.     

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 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.     

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