Inversion Of Toeplitz Matrices Research Articles

A note on the paper “A note on inversion of Toeplitz matrices” by X.-G. Lv and T.-Z. Huang

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (2007), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have given, especially when the size of the Toeplitz matrix is very large. Numerical experiments illustrate the effectiveness of our theoretical results.

Explicit formulas for the inverses of Toeplitz matrices, with applications

We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.

Entries of the inverses of large positive definite Toeplitz matrices

This is an expository paper embarking on the asymptotic behaviorof the entries of the inverses of positive definite symmetric Toeplitzmatrices as the matrix dimension goes to infinity. We consider the behaviorof the entries in neighborhoods of the four corners as well as thedensity of the distribution of the entries over all of the inverse matrix.

Inversion of block Toeplitz and Hankel matrices

Inversion of block Toeplitz and Hankel matrices

Entries of the inverses of large positive definite Toeplitz matrices

AbstractThis is an expository paper embarking on the asymptotic behavior of the entries of the inverses of positive definite symmetric Toeplitz matrices as the matrix dimension goes to infinity. We consider the behavior of the entries in neighborhoods of the four corners as well as the density of the distribution of the entries over all of the inverse matrix.

On the inverse problem of fractional Brownian motion and the inverse of infinite Toeplitz matrices

The inverse problem of fractional Brownian motion and other Gaussian processes with stationary increments involves inverting an infinite hermitian positively definite Toeplitz matrix (a matrix that has equal elements along its diagonals). The problem of inverting Toeplitz matrices is interesting on its own and has various applications in physics, signal processing, statistics, etc. A large body of literature has emerged to study this question since the seminal work of Szegö on Toeplitz forms in 1920's. In this paper we obtain, for the first time, an explicit general formula for the inverse of infinite hermitian positive definite Toeplitz matrices. Our formula is explicitly given in terms of the Szegö function associated to the spectral density of the matrix. These results are applied to the fractional Brownian motion and to m-diagonal Toeplitz matrices and we provide explicit examples.

Inverse of the covariance matrix of an MA(2) process

A component-wise formula for the inverse of the covariance matrix of an MA(2) process is provided in Shaman (1973) requiring O(n3) calculations per column. This paper simplifies his formula resulting in only O(n) calculations per column. Using a well-known result concerning the inverse of symmetric Toeplitz matrices, we show that the inverse of the covariance matrix of an MA(2) process may be written as a function of its first column, requiring only O(n) calculations to obtain instead of the O(n2) calculations required from the Levinson–Durbin algorithm [1].

Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations

Based on the Crank-Nicolson alternating direction implicit finite volume (CN-ADI-FV) discretization of two dimensional Riesz space-fractional diffusion equations (2 D RSFDEs) proposed in [1], a fast direct solver is proposed to obtain the numerical solutions. Linear systems result from the CN-ADI-FV discretization can be treated as two matrix equations with symmetric positive definite (SPD) Toeplitz coefficient matrices. Then numerical solutions, expressed as multiple matrix-vector multiplications involving the inverse of SPD Toeplitz matrices, can be effectively obtained by using Gohberg-Semencul formula (GSF), where the preconditioned conjugate gradient (PCG) method is applied as an acceleration technique. Moreover, under certain assumptions, the spectral of the preconditioned matrices of the linear system associated with the GSF are theoretically proved to be clustered around 1. Then the PCG method is verified to converge superlinearly and can be implemented with O(NlogN) operations, where N is the number of spatial unknowns in both x- and y- directions. Besides, the PCG method is used only twice in the whole solution process, and the complete solution process can be finished with O(MN2logN) operations and O(N) memory, where M is the number of temporal unknowns. Numerical experiments are given to demonstrate the efficiency of our approach.

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

AbstractIn this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

The pseudo core inverse of a companion matrix

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

Explicit inversion of Band Toeplitz matrices by discrete Fourier transform

In this paper, we give an explicit formula for the element of the inverse where is a band Toeplitz matrix with left bandwidth s and right bandwidth r. The formula involves determinants, , whose elements are the discrete Fourier transform of where f is the symbol of .

$$\mathcal O(n)$$ O ( n ) working precision inverses for symmetric tridiagonal Toeplitz matrices with $$\mathcal O(1)$$ O ( 1 ) floating point calculations

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal a on the diagonal and b on the extra diagonals (\(a, b\in \mathbb R\)). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in \(\mathcal O(n^2)\). In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If \(\vert a\vert > 2\vert b\vert \), that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of n for sufficiently large n. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses \(\mathcal O(1)\) floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver.

Time-domain approach for multi-exciter random environment test

This paper presents a time-domain method for multi-exciter random environment tests. Traditional random environment test theory has been formulated in the frequency domain, where an important step is taking the inverse of the frequency response function matrices (FRFMs). The accuracy of this inversion tends to be poor, particularly at frequencies near lightly damped resonances. The currently used control algorithms face difficulties in suppressing abnormal spectral lines caused by this inverse problem. In this paper, traditional formulations of the environment test are reformed, and the time-domain method is adopted; this results in a more precise inverse operation in environment tests. To achieve this, reference spectra are converted into time-domain response signals. The finite long driving signals are derived by the state-space method with estimated state vectors. During the process, the inverse of rank-deficient Toeplitz matrices are stabilized with truncated singular value decomposition (TSVD) to suppress all abnormally high-level components in the driving forces; thus, overall, the spectra lines produced by noise within the frequency band are filtered out. A numerical simulation of a single-axis random vibration test of a cantilever beam is conducted using the traditional frequency-domain procedure (FDP) and the proposed time-domain procedure (TDP). The response spectra generated by both procedures are tested by control algorithms, and the result shows that responses generated by the proposed TDP are more easily controlled. The conditions of stability for both the FDP and the TDP are also determined and introduced in the simulation. Moreover, a multi-axis vibration experiment further validates the effectiveness of the TDP.

Inverse and Moore-Penrose inverse of Toeplitz matrices with classical Horadam numbers

Inverse and Moore-Penrose inverse of Toeplitz matrices with classical Horadam numbers

Orthogonal polynomials with respect to a class of Fisher-Hartwig symbols and inverse of Toeplitz matrices

Using previous results we propose an asymptotic expression of the coefficients of the orthogonal polynomials on the unit circle with respect to a weight of type \(f_{\alpha _{1},\ldots ,\alpha _{M},\, \theta _{1},\ldots ,\theta _{M}}\) defined by \({ \theta \mapsto \prod \nolimits _{1\le j \le M} \vert 1 - e^{i(\theta -\theta _{j})}\vert ^{2\alpha _{j}} c}\) with \(\theta _{j}\in ]-\pi ,\pi ]\), \(\theta _{i}\ne \theta _{j}\), and \(-\frac{1}{2} < \alpha _{j}<\frac{1}{2}\) and c a sufficiently smooth regular function. As a corollary we give an asymptotic expansion of the entries of \(T_{N}^{-1}\big ( f_{\alpha _{1},\ldots ,\alpha _{M},\theta _{1},\ldots ,\theta _{M},\, \theta _{1},\ldots ,\theta _{M}}\big )\) for \(\max ( \alpha _{1},\ldots ,\alpha _{M})\in ]0, \frac{1}{2}[\).

Determinants and Inverses of Fibonacci and Lucas Skew Symmetric Toeplitz Matrices

Determinants and Inverses of Fibonacci and Lucas Skew Symmetric Toeplitz Matrices

The stability of formulae of the Gohberg–Semencul–Trench type for Moore–Penrose and group inverses of Toeplitz matrices

We present a stability analysis of Gohberg–Semencul–Trench type formulae for the Moore–Penrose and group inverses of singular Toeplitz matrices. We develop a fast algorithm for the computation of the Moore–Penrose inverse based on a Gohberg–Semencul–Trench type formula and the LSQR method. For the group inverse, the DGMRES method is used to perform the fast computation. Numerical tests show that the fast algorithms designed here are at least as good as the known Newton iteration.

Application of the partitioning method to specific Toeplitz matrices

Abstract We propose an adaptation of the partitioning method for determination of theMoore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore-Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in MATLAB, and illustrative examples are presented.

Superfast solution of linear convolutional Volterra equations using QTT approximation

We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log2n). The results of the paper are illustrated by numerical examples.

The Inverses of Block Toeplitz Matrices

We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.