quasi-Toeplitz Matrices Research Articles

On certain matrix algebras related to quasi-Toeplitz matrices

On certain matrix algebras related to quasi-Toeplitz matrices

A defect-correction algorithm for quadratic matrix equations, with applications to quasi-Toeplitz matrices

A defect correction formula for quadratic matrix equations of the kind is presented. This formula, expressed by means of an invariant subspace of a suitable pencil, allows us to introduce a modification of the Structure-preserving Doubling Algorithm (SDA), that enables refining an initial approximation to the sought solution. This modification provides substantial advantages, in terms of convergence acceleration, in the solution of equations coming from stochastic models, by choosing a stochastic matrix as the initial approximation. An application to solving random walks in the quarter plane is shown, where the coefficients are quasi-Toeplitz matrices of infinite size.

Geometric means of quasi-Toeplitz matrices

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $$A=(a_{i,j})_{i,j=1,2,\ldots }$$ of the form $$A=T(a)+E$$ , where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series $$\sum _{k=-\infty }^{\infty } a_k e^{{{\mathfrak {i}}}k t}$$ , in the sense that $$(T(a))_{i,j}=a_{j-i}$$ . If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on $$\ell ^2$$ . We prove that if $$a_1,\ldots ,a_p$$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then matrix geometric means, such as the ALM, the NBMP and the weighted mean of quasi-Toeplitz positive definite matrices associated with $$a_1,\ldots ,a_p$$ , are quasi-Toeplitz matrices associated with the function $$(a_1\cdots a_p)^{\frac{1}{p}}$$ , which differ only by the compact correction. We introduce numerical algorithms for their computation and show by numerical tests that these operator means can be effectively approximated numerically.

Approximations for the von Neumann and Rényi entropies of graphs with circulant type Laplacians

<abstract><p>In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are $ g $-circulant matrices, i.e., circulant matrices with $ g $-periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation.</p></abstract>

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.

On the exponential of semi-infinite quasi-Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued function defined for $|z|=1$, such that $\sum_{i\in\mathbb Z}|ia_i|<\infty$, and let $E=(e_{i,j})_{i,j\in\mathbb {Z}^+}$ be such that $\sum_{i,j\in\mathbb{Z}^+}|e_{i,j}|<\infty$. A semi-infinite quasi-Toeplitz matrix is a matrix of the kind $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)$, that is, $t_{i,j}=a_{j-i}$ for $i,j\in\mathbb Z^+$. We analyze theoretical and computational properties of the exponential of $A$. More specifically, it is shown that $\exp(A)=T(\exp(a))+F$ where $F=(f_{i,j})_{i,j\in\mathbb{Z}^+}$ is such that $\sum_{i,j\in\mathbb{Z}^+}|f_{i,j}|$ is finite, i.e., $\exp(A)$ is a semi-infinite quasi-Toeplitz matrix as well, and an effective algorithm for its computation is given. These results can be extended from the function $\exp(z)$ to any function $f(z)$ satisfying mild conditions, and can be applied to finite quasi-Toeplitz matrices.

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in \mathbb Z^{+}}$ , $E=(e_{i,j})_{i,j\in \mathbb Z^{+}}$ is compact and the norms $\|a\|_{_{\mathcal {W}}}={\sum }_{i\in \mathbb Z}|a_{i}|$ and $\|E\|_{2}$ are finite. These properties allow to approximate any QT matrix, within any given precision, by means of a finite number of parameters. QT matrices, equipped with the norm $\|A\|_{_{\mathcal {Q}\mathcal {T}}}=\alpha {\|a\|}_{_{\mathcal {W}}}+\|E\|_{2}$ , for $\alpha = (1+\sqrt 5)/2$ , are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes

Denote by $\mathcal{W}_1$ the set of complex valued functions of the form $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$ which are continuous on the unit circle, and such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. We call CQT matrix a quasi-Toeplitz matrix $A$, associated with a continuous symbol $a(z)\in\mathcal W_1$, of the form $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix such that $t_{i,j}=a_{j-i}$, for $i,j\in\mathbb Z^+$, and $E=(e_{i,j})_{i,j\in\mathbb{Z}^+}$ is a semi-infinite matrix such that $\sum_{i,j=1}^{+\infty}|e_{i,j}|$ is finite. We prove that the class of CQT matrices is a Banach algebra with a suitable sub-multiplicative matrix norm $\|\cdot\|$. We introduce a finite representation of CQT matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind $AX^2+BX+C=0$, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where $A,B,C$ are CQT matrices.

Exact solution of corner-modified banded block-Toeplitz eigensystems

Motivated by the challenge of seeking a rigorous foundation for the bulk-boundary correspondence for free fermions, we introduce an algorithm for determining exactly the spectrum and a generalized-eigenvector basis of a class of banded block quasi-Toeplitz matrices that we call corner-modified. Corner modifications of otherwise arbitrary banded block-Toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance. Our algorithm leverages the interplay between a non-standard, projector-based method of kernel determination (physically, a bulk-boundary separation) and families of linear representations of the algebra of matrix Laurent polynomials. Thanks to the fact that these representations act on infinite-dimensional carrier spaces in which translation symmetry is restored, it becomes possible to determine the eigensystem of an auxiliary projected block-Laurent matrix. This results in an analytic eigenvector Ansatz, independent of the system size, which we prove is guaranteed to contain the full solution of the original finite-dimensional problem. The actual solution is then obtained by imposing compatibility with a boundary matrix, whose shape is also independent of system size. As an application, we show analytically that eigenvectors of short-ranged fermionic tight-binding models may display power-law corrections to exponential behavior, and demonstrate the phenomenon for the paradigmatic Majorana chain of Kitaev.

Spectral Analysis of Nonsymmetric Quasi-Toeplitz matrices with Applications to Preconditioned Multistep Formulas

The eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time-dependent partial differential equations is analyzed and discussed. The matrices generated by the underlying numerical integrators are small rank perturbations of block Toeplitz matrices; circulant-like preconditioners based on the former are considered. The eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of Krylov iterative accelerators. Due to several reasons (lack of symmetry, band structure, and coefficients depending on the size) the classical approach based on smooth generating functions gives very little insight here. Therefore, to characterize the eigenvalues, a difference equation approach exploiting the band Toeplitz and circulant patterns generalizing the well-known results of Trench is proposed.

Problems with the toeplitz and quasi-toeplitz matrices that appear in the static stability analysis of elastic systems

Three stability problems are considered for layered elastic systems. The problems can be treated completely analytically. These systems are packs consisting of arbitrary numbers of identical infinite plates attached by identical layers of a soft medium modelled as a Winkler foundation. The problems differ in the nature of the boundary conditions assigned to the faces of the packets. Each face can be either free, or attached to a non-deformable support. A cylindrical corrugation is sought everywhere. In all cases the critical force, the whole load spectrum and all the forms of stability loss are given by simple explicit formulae.

The Asymptotic Spectra of Banded Toeplitz and Quasi-Toeplitz Matrices

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Gohberg-Semencul type formulas via embedding of Lyapunov equations (signal processing)

The authors present a new way of deriving Gohberg-Semencul-type inversion formulas for Hermitian Toeplitz and quasi-Toeplitz matrices. The approach is based on a certain Sigma -lossless embedding of Lyapunov equations. It has been shown that if a nonsingular matrix R has Toeplitz displacement inertia (p, q), then R/sup -1/ does not have the same Toeplitz displacement inertia. However, a para-Hermitian conjugate of R/sup -1/ will have this property. It is also shown that the Gohberg-Semencul-type inversion formulas can be formed directly in terms of certain parameters of the embedding. >

Immitance-Type Three-Term Schur and Levinson Recursions for Quasi-Toeplitz Complex Hermitian Matrices

A comprehensive analysis is made of Schur- and Levinson-type algorithms for Toeplitz and quasi-Toeplitz matrices that have half the number of multiplications and the same number of additions as the classical algorithms. Several results of this type have appeared in the literature under the label “split algorithms.” In this approach the reduction in computation is obtained by a two-step procedure: (i) the first step is a variable (or “recursion-type” ) transformation from the classical (i.e., “scattering” ) variables to a new (so-called, “immitance”) set of variables, which by itself reduces the number of multiplications at the cost of increasing the number of additions; (ii) the second step achieves control of the number of additions by converting the two-term recursions into the lesser known (for discrete orthogonal polynomials) three-term recursions. In the Toeplitz case the new variables turn out to be the odd and even parts of the classical variables, leading to the terminology of split algorithms, bu...

Immittance- versus scattering-domain fast algorithms for non-Hermitian Toeplitz and quasi-Toeplitz matrices

The classical algorithms of Schur and Levinson are efficient procedures to solve sets of Hermitian Toeplitz linear equations or to invert the corresponding coefficient matrices. They propagate pairs of variables that may describe incident and scattered waves in an associated cascade-of-layered-media model, and thus they can be viewed as scattering-domain algorithms. It was recently found that a certain transformation of these variables followed by a change from two-term to three-term recursions results in reduction in computational complexity in the abovementioned algorithms roughly by a factor of two. The ratio of such pairs of transformed variables can be interpreted in the above layered-media model as an impedance or ad mittance; hence the name immittance-domain variables. This paper provides extensions for previous immittance Schur and Levinson algorithms from Hermitian to non-Hermitian matrices. It considers both Toeplitz and quasi-Toeplitz matrices (matrices with certain “hidden” Toeplitz structure) and compares two- and three-term recursion algorithms in the two domains. The comparison reveals that for non-Hermitian matrices the algorithms are equally efficient in both domains. This observation adds new comprehension to the source and value of algorithms in the immittance domain. The immittance algorithms, like the scattering algorithms, exploit the (quasi-)Toeplitz structure to produce fast algorithms. However, unlike the scattering algorithms, they can respond also to symmetry of the underlying matrix when such extra structure is present, and yield algorithms with improved efficiency.

Immittance-domain Levinson algorithms

Several computationally efficient versions of the Levinson algorithm for solving linear equations with Toeplitz and quasi-Toeplitz matrices are presented, motivated by a new stability test. The new versions require half the number of multiplications and the same number of additions as the conventional form of the Levinson algorithm. The saving is achieved by using three-term (rather than two-term) recursions and propagating them in an impedance/admittance (or immittance) domain rather than the conventional scattering domain. One of the recursions coincides with recent results of P. Delsarte and Y. Genin (IEEE Trans., Acoust. Speech, Signal Proc., vol.ASSP-34, p.470-8, June 1986) on split Levinson algorithms for symmetric Toeplitz matrices, where the efficiency is gained by using the symmetric and skew-symmetric versions of the usual polynomials. This special structure is lost in the quasi-Toeplitz case, but one still can obtain similar computational reductions by suitably using three-term recursions in the immittance domain.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Inversion and factorization of non-Hermitian quasi-Toeplitz matrices

This paper considers formulas and fast algorithms for the inversion and factorization of non-Hermitian Toeplitz and quasi-Toeplitz (QT) matrices (matrices with a certain “hidden” Toeplitz structure). The results include the following generalizations: (1) A Schur algorithm that extends to non-Hermitian matrices a previous triangular factorization algorithm for Hermitian QT matrices. (2) A Levinson algorithm that generalizes to non-Hermitian matrices a previous Levinson algorithm that finds the triangularly factorized inverses of certain (so-called admissible) QT matrices. (3) The extension to QT matrices of the Gohberg-Semencul (GS) inversion formula for non-Hermitian Toeplitz matrices. Next, the paper introduces a new fast algorithm, called the extended QT factorization algorithm, that overcomes the restriction to admissibility matrices of the above Levinson algorithm. The new algorithm is efficient and comprehensive; it produces, for a general QT matrix R n of size ( n + 1)×( n + 1), the triangularly factorized inverses and the GS type inverses of the matrix and all its submatrices, as well as the triangular factorization of R n itself, all in approximately 7 n 2 elementary operations for a non-Hermitian and 3.5 n 2 for a Hermitian matrix. The fast algorithms for non-Hermitian QT matrices are shown to be associated with two discrete transmission lines (which reduce to the familiar single lattice in the Hermitian case). All the presented algorithms are illustrated and interpreted in terms of input sequences and flows of signals in the related transmission line realization.