In this paper, a so-called auxiliary matrix polynomial Xn(z) and a true right formal orthogonal matrix polynomial (FOMP) A 1 n (z) is connected to each well-conditioned leading principal block submatrix of a given block Toeplitz matrix. From these two matrix polynomi-als, all other right FOMPs of block n of a system of block biorthogonal matrix polynomials with respect to the block Toeplitz mom...

A Toeplitz matrix has constant diagonals; a multilevel Toeplitz matrix is defined recursively with respect to the levels by replacing the matrix elements with Toeplitz blocks. Multilevel Toeplitz linear systems appear in a wide range of applications in science and engineering. This paper discusses an MPI implementation for solving such a linear system by using the conjugate gradient algorithm. ...

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks ...

Many issues in signal processing involve the inverses of Toeplitz matrices. One widely used technique is to replace Toeplitz matrices with their associated circulant matrices, based on the well-known fact that Toeplitz matrices asymptotically converge to their associated circulant matrices in the weak sense. This often leads to considerable simplification. However, it is well known that such a ...

In the rst part 13] of the paper transformationsmappingToeplitz and Toeplitz-plus-Hankel matrices into generalizedCauchy matrices were studied. In this second part fast algorithms for LU-factorization and inversion of generalized Cauchy matrices are discussed. It is shown that the combinationof transformation pivoting techniques leads to algorithms for indeenite Toeplitz and Toeplitz-plus-Hanke...

In this paper two fast algorithms that use orthogonal similarity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form are developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost efficiency by exploiting the rank structure of the input Toeplitz matrix. The proposed algorithms differ in the choice...

This paper presents two universal algorithms for generalized discrete matrix Bellman equations with symmetric Toeplitz matrix. The algorithms are semiring extensions of two well-known methods solving Toeplitz systems in the ordinary linear algebra.

The problem of computing an approximate solution of an overdetermined system of linear equations is considered. The usual approach to the problem is least squares, in which the 2-norm of the residual is minimized. This produces the minimum variance unbiased estimator of the solution when the errors in the observations are independent and normally distributed with mean 0 and constant variance. I...

It is well known that the generating function f ∈ L([−π, π],R) of a class of Hermitian Toeplitz matrices An(f) describes very precisely the spectrum of each matrix of the class [U. Grenader and G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984; E. E. Tyrtyshnikov, Linear Algebra Appl., 232 (1996), pp. 1–43]. In this paper we consider n×n block Toeplitz matrices w...

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp. multi-T...