Tensor product factorization-based numerical algorithms for the inverses of generalized banded Toeplitz matrices

In compressive sensing (CS), the restricted isometry property (RIP) is an important condition on measurement matrices which guarantees the recovery of sparse signals with undersampled measurements. It has been proved in the prior works that both random (e.g., i.i.d. Gaussian, Bernoulli, $\ldots$ ) and Toeplitz matrices satisfy the RIP with high probability. However, structured matrices, such as banded Toeplitz matrices have drawn more attention since their structures have the advantage of fast matrix multiplication which may decrease the computational complexity of recovery algorithms. In this paper, we show that banded block Toeplitz matrices satisfy the RIP condition with high probability. Banded block Toeplitz matrices can be used in the sparse multi-channel source separation. The banded block Toeplitz matrices decrease the computational complexity while they have fewer number of non-zero entries in comparison to the same dimensional banded Toeplitz matrices. Furthermore, our simulation results show that banded block Toeplitz matrices outperform banded Toeplitz matrices in signal estimation. The analytical RIP bound for banded block Toeplitz matrices is provided in this paper and the RIP bound of sparse Gaussian matrices is also obtained as an upper bound for banded block Toeplitz matrices. Our simulation and analytical results show that sparse Gaussian random matrices do satisfy the RIP condition with high probability. The probability of satisfying the RIP depends on the probability of zero entries.

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue.

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

Arithmetic complexity and numerical properties of algorithms involving Toeplitz matrices

For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.

Banded Toeplitz matrices of large size occur in many practical problems [1]-[6]. Here the problem of inversion as well as the problem of solving simultaneous equations of the type Hx = y, when H is a large banded Toeplitz matrix, are considered. It is shown via certain circular decompositions of H that such equations may be exactly solved in O(N \log_{2} N) rather than in O(N2) computations as in Levinson-Trench algorithms. Furthermore, the algorithms of this paper are nonrecursive (as compared to the Levinson-Trench algorithms), and afford parallel processor architectures and others such as transversal filters [17] where the computation time becomes proportional to N rather than to N \log N . Finally, a principle of matrix decomposition for fast inversion of matrices is introduced as a generalization of the philosophy of this paper.

The authors consider the solutions of Hermitian Toeplitz-plus-band systems $(A_n + B_n )x = b$, where $A_n $ are n-by-n Toeplitz matrices and $B_n $ are n-by-n band matrices with bandwidth independent of n. These systems appear in solving integrodifferential equations and signal processing. However, unlike the case of Toeplitz systems, no fast direct solvers have been developed for solving them. In this paper, the preconditioned conjugate gradient method with band matrices as preconditioners is used. The authors prove that if $A_n $ is generated by a nonnegative piecewise continuous function and $B_n $ is positive semidefinite, then there exists a band matrix $C_n $, with bandwidth independent of n, such that the spectra of $C_n^{ - 1} (A_n + B_n )$ are uniformly bounded by a constant independent of n. In particular, we show that the solution of $(A_n + B_n )x = b$ can be obtained in $O(n\log n)$ operations.

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.

The aim of this contribution is to show that the performance of the recently developed high performance algorithm for evaluating linear recursive filters can be increased by using new generalized data structures for dense matrices introduced by F. G. Gustavson. The new implementation is based on vectorized algorithms for banded triangular Toeplitz matrix - vector multiplication and the algorithm for solving linear recurrence systems with constant coefficients. The results of experiments performed on Intel Itanium 2 and Cray X1 are also presented and discussed.

It is well known that the mutual coupling between array elements is inversely related to their separation and may be negligible for elements separated by a few wavelengths, and at the same time the coupling between any two equally spaced sensors is approximately the same. Moreover, following the principle of reciprocity, the mutual coupling can always be expressed by a symmetric matrix. Consequently, a banded symmetric Toeplitz matrix provides a satisfactory model for the mutual coupling of ULA (Uniform linear array). In this paper, a robust subspace-based direction-of-arrival (DOA) estimation algorithm in the presence of mutual coupling is proposed for ULA. Based on the banded symmetric Toeplitz matrix model for the mutual coupling of ULA, the new algorithm provides a favorable DOA estimates and exploits no knowledge of the sensor mutual couplings. More importantly, an accurate estimate of mutual coupling matrix can also be achieved simultaneously for array calibration.

The authors propose a technique to compute signal bearings for the situation when the desired signal is corrupted by additive noise with unknown covariance. The noise covariance matrix is assumed to be a banded symmetric Toeplitz matrix. This algorithm eliminates the need for the evaluation of eigenvalues and eigenvectors. The algorithm is also computationally very efficient, since it requires the use of a simpler orthogonal decomposition, i.e. only the Householder decomposition. The number of computations required to perform this decomposition is of the order of O(2N/sup 3//3), while the eigendecomposition requires O(N/sup 4/) computations. >

BANDED MATRIX SOLVERS AND POLYNOMIAL DIOPHANTINE EQUATIONS

For a family of near banded Toeplitz matrices, generalized characteristic polynomials are shown to be orthogonal polynomials of two variables, which include the Chebyshev polynomials of the second kind on the deltoid as a special case. These orthogonal polynomials possess a maximal number of real common zeros, which generate a family of Gaussian cubature rules in two variables.