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.     

On Some Properties of Polynomial Bases of Subspaces over the Field of Rational Functions in Several Variables

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a free polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.     

State-Space Minimal Realizations and Latent Structures of Column-Reduced and Nonsingular Polynomial Matrices

The concept of a column-reduced polynomial matrix is an importantone in the theory of linear systems. The theory of Jordan chainsand minimal realizations is developed for such matrices. Also,the relationships between generalized latent vectors of thenonsingular polynomial matrix and their associated generalizedeigenvectors of the system map are explored in this paper. Thispermits the spectral analysis of an arbitrary nonsingular polynomialmatrix, extending previous work for the monic case.     

Hyman’s method revisited

The QR algorithm is considered one of the most reliable methods for computing matrix eigenpairs. However, it is unable to detect multiple eigenvalues and Jordan blocks. Matlab’s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors. This paper calls attention to Hyman’s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity, an important information for more reliable computation. We suggest a test matrix collection for Jordan blocks. The first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work.     

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 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.     

线性定常系统非齐次两点边值问题的扩展精细积分方法

提出了一种求解非齐次线性两点边值问题的高精度和高稳定的扩展精细积分方法（EPIM）.首先引入了区段量（即区段矩阵和区段向量）来离散非齐次线性微分方程,建立了非齐次两点边值问题基于区段量的求解框架.在该框架下,不同区段的区段量可以并行计算,整体代数方程组的集成不依赖于边界条件.然后引入区段响应矩阵来处理两点边值问题的非齐次项,导出了多项式函数、指数函数、正/余弦函数及其组合函数形式的非齐次项对应的区段响应矩阵的加法定理,结合增量存储技术提出了EPIM.对具有上述函数形式的非齐次项,该方法可以得到计算机上的精确解,一般形式的非齐次项则利用上述函数近似求解.最后通过两个具有刚性特征的数值算例验证了该方法的高精度和高稳定性.     

To Solving Multiparameter Problems of Algebra. I. Methods for Computing Complete Polynomials and Their Applications

The notion of a complete polynomial of a multiparameter polynomial matrix, generalizing that of an invariant polynomial of a one-parameter polynomial matrix, is introduced. Methods for computing complete polynomials are suggested. Applications of these methods to various spectral problems for polynomial matrices are considered. Bibliography: 7 titles.     

Computing the Invariant Polynomials of a Polynomial Matrix. II

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the W and V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles.     

A Method of computing a fundamental row of solutions and jordan chains for a singular linear pencil of matrices

An algorithm is suggested for constructing a fundamental row of polynomial solutions for a singular linear pencil of matrices. The algorithm is a modification of the algorithm for constructing polynomial solutions of a pencil that was suggested earlier by the author. For a computed eigenvalue of the regular kernel of the singular pencil, corresponding chains of Jordan vectors are found. In the process, essential use is made of the subspace generated by a fundamental row of polynomial solutions for the eigenvalue under consideration.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 101–113, 1983.     

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 spectral problems for polynomial matrices

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.     

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.     

利用矩阵的初等变换求方阵的特征值

高阶方阵的特征多项式以及特征值的求得,在计算上往往有一定的难度.本文首先从理论上分析了存在一个上三角矩阵或者下三角矩阵与一个方阵相似;接着,提出了相似变换的概念,分析了相似变换中初等矩阵的选择方法;然后指出了利用相似变换在求方阵的特征多项式以及特征值时的方法,并列举若干实例给予了说明.     

一种求三维弹性力学基本解析解的特征方程解法

提出了一种简单的推导各向同性材料,三维弹性力学问题基本解析解的特征方程解法．应用三维问题控制微分方程的算子矩阵,通过计算其行列式可得到问题特征通解所需满足的特征方程．将满足各种不同简化特征方程的特征通解,代入到微分方程算子矩阵所对应的不同的缩减伴随矩阵,可推导得出相应的三维弹性力学问题的基本解析解,包括B-G解、修正的P-N（P-N-W）解和类胡海昌解．进一步对各类多项式形式的基本解析解的独立性进行了讨论．这些工作为构造数值方法中所需的完备独立的解析试函数奠定了基础．     

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.     

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.     

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

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.     

Multivariate subresultants using Jouanolou matrices

Earlier results expressing multivariate subresultants as ratios of two subdeterminants of the Macaulay matrix are extended to Jouanolou matrices. These matrix constructions are generalizations of the classical Macaulay matrices and involve matrices of significantly smaller size. Equivalence of the various subresultant constructions is proved. The resulting subresultant method improves the efficiency of previous methods to compute the solution of over-determined polynomial systems.