Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.     

Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

Fast Structured Matrix Computations: Tensor Rank and Cohn–Umans Method

We discuss a generalization of the Cohn–Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn–Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen’s tensor rank approach, the traditional framework for investigating bilinear complexity. To demonstrate the utility of the generalized method, we apply it to find the fastest algorithms for forming structured matrix–vector product, the basic operation underlying iterative algorithms for structured matrices. The structures we study include Toeplitz, Hankel, circulant, symmetric, skew-symmetric, f-circulant, block Toeplitz–Toeplitz block, triangular Toeplitz matrices, Toeplitz-plus-Hankel, sparse/banded/triangular. Except for the case of skew-symmetric matrices, for which we have only upper bounds, the algorithms derived using the generalized Cohn–Umans method in all other instances are the fastest possible in the sense of having minimum bilinear complexity. We also apply this framework to a few other bilinear operations including matrix–matrix, commutator, simultaneous matrix products, and briefly discuss the relation between tensor nuclear norm and numerical stability.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Using FFT-based techniques in polynomial and matrix computations: recent advances and applicatons

Computations with univariate polynomials, like the evaluation of product, quotient, remainder, greatest common divisor, etc, are closely related to linear algebra computations performed with structured matrices having the Toeplitz-like or the Hankel-like structures.

The discrete Fourier transform, and the FFT algorithms for its computation, constitute a powerful tool for the design and analysis of fast algorithms for solving problems involving polynomials and structured matrices.

We recall the main correlations between polynomial and matrix computations and present two recent results in this field: in particular, we show how Fourier methods can speed up the solution of a wide class of problems arising in queueing theory where certain Markov chains, defined by infinite block Toeplitz matrices in generalized Hessenberg form, must be solved. Moreover, we present a new method for the numerical factorization of polynomials based on a matrix generalization of Koenig's theorem. This method, that relies on the evaluation/interpolation technique at the Fourier points, reduces the problem of polynomial factorization to the computation of the LU decomposition of a banded Toeplitz matrix with its rows and columns suitably permuted. Numerical experiments that show the effectiveness of our algorithms are presented     

Extremal Trigonometric and Power Polynomials in Several Variables

We study extremal nonnegative polynomials in several variables. Our approach makes substantial use of block Toeplitz matrices. Note that the blocks of these matrices are themselves Toeplitz matrices.     

Kernel structure of Toeplitz-plus-Hankel matrices

The structure of the kernel of block Toeplitz-plus-Hankel matrices R=[a_j−k+b_j+k], where a_j and b_j are the given p×q blocks with entries from a given field, is investigated. It is shown that R corresponds to two systems of at most p+q vector polynomials from which a basis of the kernel of R and all other Toeplitz-plus-Hankel matrices with the same parameters a_j and b_j can be built. The main result is an analogue of a known kernel structure theorem for block Toeplitz and block Hankel matrices.     

On inversion of block Toeplitz matrices

In [1] we proved that each inverse of a Toeplitz matrix can be constructed via three of its columns, and thus, a parametrization of the set of inverses of Toeplitz matrices was obtained. A generalization of these results to block Toeplitz matrices is the main aim of this paper.     

Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that our multigrid solvers lead to optimal order convergence. AMS subject classification (2000) 65N55, 65F10     

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.     

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [−1, 1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.     

A New Inertia Theorem for Stein Equations,Inertia of Invertible Hermitian Block Toeplitz Matrices and Matrix Orthogonal Polynomials

In this paper we present an inertia result for Stein equations with an indefinite right hand side. This result is applied to establish connnections between the inertia of invertible hermitian block Toeplitz matrices and associated orthogonal polynomials.     

On regularity of block triangular fuzzy matrices

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relational equations consistency is studied.     

Band Toeplitz preconditioners for block Toeplitz systems

We consider the solutions of block Toeplitz systems with Toeplitz blocks by the preconditioned conjugate gradient (PCG) method. Here the block Toeplitz matrices are generated by nonnegative functions f(x,y). We use band Toeplitz matrices as preconditioners. The generating functions g(x,y) of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing (f − g)/f∞. We prove that the condition number of the preconditioned system is O(1). An a priori bound on the number of iterations for convergence is obtained.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

The Bezoutian and the algebraic Riccati equation

The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.     

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

In this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices.     

Resultants and group matrices

In this paper, we study the resultants of polynomials which are the determinants of polynomial matrices, and we show that the determinant of a group matrix can be expressed in terms of an iterated resultant of a system of polynomials in several variables which are explicitly given in terms of first row entries. We also show how to obtain the block triangulation of block group matrices over a finite field.     

Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.     

Block LU-factorization of confluent Vandermonde matrices

In this note we show that an asymptotically fast algorithm may be designed in order to realize a block LU-factorization of confluent Vandermonde matrices. This result is based on a displacement structure satisfied by confluent Vandermonde matrices and on factorizations of the block elements in terms of triangular Toeplitz matrices.