On unitary interpolants and Fredholm infinite block Toeplitz matrices

Conditions for the existence and uniqueness of unitary n × n matrix valued functions f on the unit circle with prescribed Fourier coefficients f_j for j 0 are given (in terms of infinite block Hankel matrices based on the prescribed coefficients f₀,f₁, ) for a natural class of functions. A unitary function belongs to this class if and only if it admits a generalized factorization (in a sense which will be made precise in the paper) or equivalently if and only if any one (and hence both) of the two Toeplitz operators defined by the function are Fredholm. In particular this class includes all continuous unitary n × n matrix valued functions. It is shown that the nonnegative factorization indices of every such unitary f are uniquely determined by f₀,f₁, and formulas for them are given.     

Hankel matrices and the infinite companion

A formula of Barnett type relating the Bezoutian B(f,g) to the Hankel matrix H(g/f) is extended to rectangular Bezoutians. The proof is based on an interesting relation between the family of all Hankel matrices corresponding to the Markov parameters of g/f and the infinite companion matrix corresponding to f.     

Incomplete block factorization methods for complex-structure matrices

Two classes of SSOR-type incomplete block factorization methods are proposed for preconditioning of linear algebraic systems of equations with block banded matrices of complex structure. Correctness conditions are derived for these methods in application to M-matrices and their efficiency is demonstrated by numerical experiments with linear algebraic systems obtained by discretization of the three-dimensional Poisson equation using quadratic and cubic serendipity finite elements. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 5–22, 1987.     

A block Cholesky-LU-based QR factorization for rectangular matrices

The Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block-partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well-known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.     

Invariant factors, Wiener-Hopf factorization indices, and invariant factors at infinity of rational matrices

The relationship between the finite structure, the infinite structure, and the Wiener-Hopf factorization indices of any rectangular rational matrix is studied.     

Hankel and Loewner matrices

Mutual relations between the Hankel, Toeplitz, Bézout, and Loewner matrices as well as further connections to rational interpolation and projective geometry are investigated.     

Polynomials and Hankel matrices

Compatibility of a Hankel n × n matrix H and a polynomial f of degree m, m ? n, is defined. If m = n, compatibility means that HC^T_f=C_fH where C_f is t companion matrix of f. With a suitable generalization of C_f, this theorem is generalized to the case that m < n.     

Hankel integral operators and isometric interpolants on the line

The problem of finding an m × n continuous isometric matrix valued function with entries from the Wiener algebra on the line and with prespecified Fourier inverse transform on the half line is studied. Conditions for existence, uniqueness, and formulas for the solution of this problem are presented. Connections are made between the positive factorization indices of certain solutions to this problem and the dimensions of the kernels of Hankel operators based on the prespecified data alluded to above. The paper uses techniques and results developed in an earlier study of an interpolation problem on the circle. The main theorems are in fact continuous analogues of the latter.     

On Hermitian block Hankel matrices,matrix polynomials,the Hamburger moment problem,interpolation and maximum entropy

Reproducing kernel space methods are used to study the truncated matrix Hamburger moment problem on the line, an associated interpolation problem and the maximum entropy solution. Enroute a number of formulas are developed for orthogonal matrix polynomials associated with a block Hankel matrix (based on the specified matrix moments for the Hamburger problem) under less restrictive conditions than positive definiteness. An analogue of a recent formula of Alpay-Gohberg and Gohberg-Lerer for the number of roots of certain associated matrix polynomials is also established.The author would like to acknowledge with thanks Renee and Jay Weiss for endowing the chair which supported this research.     

Hankel matrices and quadratic forms

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.     

Block LU factorization is stable for block matrices whose inverses are block diagonally dominant

Let A M_n (C) and let the inverse matrix B = A^–1 be block diagonally dominant by rows (columns) w.r.t. an m × m block partitioning and a matrix norm. We show that A possesses a block LU factorization w.r.t. the same block partitioning, and the growth factor for A in this factorization is bounded above by 1 + , where = max _{1im i} and _i, 0 _i 1, are the row (column) block dominance factors of B. Further, the off-diagonal blocks of A (and of its block Schur complements) satisfy the inequalities

Componentwise error analysis for the block LU factorization of totally nonnegative matrices

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.