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.     

The least-squares method for matrices dependent on parameters

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.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

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.     

The vector-valued Wiener-Hopf equation for a mixed boundary-value problem of diffraction at a wedge intersected by a half-plane

For a mixed boundary-value problem in a nonregular region we obtain the vector-valued Wiener-Hopf equation, which is then reduced to infinite systems of linear algebraic equations using the factorization method and Liouville's theorem. It then becomes possible to solve the equation with prescribed precision for arbitrary values of the parameters of the problem. In the stationary case the solution is obtained in closed form.Translated fromMatematichni Metodi ta Fiziko-Mechanichni Polya, Vol. 40, No. 3, 1997, pp. 87–92.     

Spectral optimization of explicit iterative methods. I

Methods of constructing preconditioning of explicit iterative methods of solving systems of linear, algebraic equations with sparse matrices are considered in the work. The techniques considered can, first of all, be realized within the framework of the simplest data structures; secondly, the graph structures of the corresponding algorithms are well adapted to realization on parallel computers; thirdly, in conjunction with modifications of Chebyshev methods they make it possible to construct rather effective computational algorithms. Experimental data are presented which demonstrate the effect of the proposed techniques of preconditioning of the distribution of the eigenvalues of matrices of systems arising in discretization of two-dimensional elliptic boundary-value problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 51–60, 1984.     

On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW() which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.     

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.     

On parallel solution of linear elasticity problems. Part II: Methods and some computer experiments

This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block‐diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg‐method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M‐matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block‐diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)‐factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher‐order finite elements. Copyright © 2002 John Wiley & Sons, Ltd.     

A backward stability analysis of diagonal pivoting methods for solving unsymmetric tridiagonal systems without interchanges

This paper concerns the LBM ^T factorization of unsymmetric tridiagonal matrices, where L and M are unit lower triangular matrices and B is block diagonal with 1×1 and 2×2 blocks. In some applications, it is necessary to form this factorization without row or column interchanges while the tridiagonal matrix is formed. Bunch and Kaufman proposed a pivoting strategy without interchanges specifically for symmetric tridiagonal matrices, and more recently, Bunch and Marcia proposed pivoting strategies that are normwise backward stable for linear systems involving such matrices. In this paper, we extend these strategies to the unsymmetric tridiagonal case and demonstrate that the proposed methods both exhibit bounded growth factors and are normwise backward stable. Copyright © 2009 John Wiley & Sons, Ltd.     

Massively parallel preconditioners for symmetric positive definite linear systems

In this paper, we address the problem of solving sparse symmetric linear systems on parallel computers. With further restrictive assumptions on the matrix (e.g., bidiagonal or tridiagonal structure), several direct methods may be used. These methods give ideas for constructing efficient data parallel preconditioners for general positive definite symmetric matrices. We describe two examples of such preconditioners for which the factorization (i.e., the construction of the preconditioning matrix) turns out to be parallel. This revised version was published online in June 2006 with corrections to the Cover Date.     

An algebraic multifrontal preconditioner that exploits the low‐rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.     

Sparse direct factorizations through unassembled hyper-matrices

We set out to efficiently compute the solution of a sequence of linear systems A_ix_i = b_i, where the matrix A_i is tightly related to matrix A_{i –1}. In the setting of an hp -adaptive Finite Element Method, the sequence of matrices A_i results from successive local refinements of the problem domain. At any step i > 1, a factorization already exists and it is the updated linear system relative to the refined mesh for which a factorization must be computed in the least amount of time. This observation holds the promise of a tremendous reduction in the cost of an individual refinement step. We argue that traditional matrix storage schemes, whether dense or sparse, are a bottleneck, limiting the potential efficiency of the solvers. We propose a new hierarchical data structure, the Unassembled Hyper-Matrix (UHM), which allows the matrix to be stored as a tree of unassembled element matrices, hierarchically ordered to mirror the refinement history of the physical domain. The factorization of such an UHM proceeds in terms of element matrices, only assembling nodes when they need to be eliminated. Efficiency comes in terms of both performance and space requirements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constructive scattering theory

We consider a problem of factoring the scattering matrix for Schrödinger equation on the real axis. We find the elementary factorization blocks in both the finite and infinite cases and establish a relation to the matrix conjugation problem. We indicate a general scheme for constructing a large class of scattering matrices admitting a quasirational factorization.     

Symmetric Permutations for I-Matrices to Avoid Small Pivots During Incomplete Factorization

Incomplete LU–factorizations have been very successful as preconditioners for solving sparse linear systems iteratively. However, for unsymmetric, indefinite systems small pivots (or even zero pivots) are often very detrimental to the quality of the preconditioner. A fairly recent strategy to deal with this problem has been to permute the rows of the matrix and to scale rows and columns to produce an I–matrix, a matrix having elements of modulus one on the diagonal and elements of at most modulus one elsewhere. These matrices are generally more suited for incomplete LU–factorization. I–matrices are preserved by symmetric permutation, i.e. by applying the same permutation to rows and columns of a matrix. We discuss different approaches for constructing such permutations which aim at improving the sparsity and diagonal dominance of an initial block. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary-Value problems for systems of integro-differential equations with Degenerate Kernel

By using methods of the theory of generalized inverse matrices, we establish a criterion of solvability and study the structure of the set of solutions of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel.     

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

A direct and inverse scattering theory on the full line is developed for a class of first-order selfadjoint 2n×2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.Dedicated to Israel Gohberg on the Occasion of his 70^th Birthday     

Bilus: A block version of ilus factorization

ILUS factorization has many desirable properties such as its amenability to the skyline format, the ease with which stability may be monitored, and the possibility of constructing a preconditioner with symmetric structure. In this paper we introduce a new preconditioning technique for general sparse linear systems based on the ILUS factorization strategy. The resulting preconditioner has the same properties as the ILUS preconditioner. Some theoretical properties of the new preconditioner are discussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new preconditioner is cheaper to construct than the ILUS preconditioner.     

Minimal factorizations of rational matrix functions in terms of polynomial models

The problem of minimal factorization of rational bicausal matrices is considered, using polynomial models. The factors are given explicity in terms of polynomial matrices.