Positive Definite Toeplitz Matrix Research Articles

On the banded Toeplitz structured distance to symmetric positive semidefiniteness

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.

Modulus-based circulant and skew-circulant splitting iteration method for the linear complementarity problem with a Toeplitz matrix

By reformulating the linear complementarity problem involving a positive definite Toeplitz matrix as an equivalent fixed-point system, we construct a modulus-based circulant and skew-circulant splitting (MCSCS) iteration method. We also analyze the convergence of the method and show that the new method is effective by providing some numerical results.

A circulant preconditioner for the Riesz distributed-order space-fractional diffusion equations

ABSTRACT In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix is proved to be clustered around 1, which can guarantee the superlinear convergence rate of the proposed method. Numerical experiments are carried out to demonstrate the effectiveness of our proposed method.

A preconditioned modulus-based matrix multisplitting block iteration method for the linear complementarity problems with Toeplitz matrix

A preconditioned modulus-based matrix multisplitting block iteration method is presented for solving the linear complementarity problem with symmetric positive definite Toeplitz matrix. We choose Strang’s preconditioner or T. Chan’s preconditioner as preconditioner in the method. The method has faster convergence rate and less computational work. We also analyze convergence of the method, and show that the new method is effective with some numerical results.

Wideband Spectrum Sensing via Derived Correlation Matrix Completion Based on Generalized Coprime Sampling

Wideband spectrum sensing is a popular topic in signal processing, especially for many radar and communication applications. What we face is a high sampling rate and a large volume of samples, in which demand of reducing the sampling rate without sacrificing the sensing resolution and quality. The generalized coprime sampling can break the limitation of the Nyquist sampling theorem with both characteristics of sparse sensing and coprime numbers. To fully utilize all the information received of the derived correlation matrix constructed by the different time delays, the matrix completion method is exploited. The theory of matrix completion is an extension of compressive sensing, though, which is not restrained by the sparsity and the restricted isometry property. The interpolation-based method presented via the convex framework of the nuclear norm minimization has no extra fine-tuned parameters, which different from techniques like compressive covariance sampling, positive definite Toeplitz matrix completion, and so on. Moreover, compared to the selection-based method under a continuous set, the proposed method improves the spectral resolution and estimation accuracy to avoid the information losing. The Simulation results indicate the performance of the algorithm.

Data-driven spectral analysis of the Koopman operator

Abstract Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel–Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an N × N Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

The Bauer-Type Factorization of Matrix Polynomials Revisited and Extended

For a Laurent polynomial $$a(\lambda )$$ , which is Hermitian and positive definite on the unit circle, the Bauer method provides the spectral factorization $$a(\lambda ) = p(\lambda )p{\kern 1pt} {\text{*}}({{\lambda }^{{ - 1}}})$$ , where $$p(\lambda )$$ is a polynomial having all its roots outside the unit circle. Namely, as the size of the banded Hermitian positive definite Toeplitz matrix associated with the Laurent polynomial increases, the coefficients at the bottom of its Cholesky lower triangular factor tend to the coefficients of $$p(\lambda )$$ . We study extensions of the Bauer method to the non-Hermitian matrix case. In the Hermitian case, we give new convergence bounds with computable coefficients.

The Circulant Rational Covariance Extension Problem: The Complete Solution

The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle rather than on the discrete real line . The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.

Generalized circulant Strang‐type preconditioners

SUMMARYStrang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {eiφ}‐circulant Strang‐type preconditioner. Copyright © 2011 John Wiley & Sons, Ltd.

New Square-Root Factorization of Inverse Toeplitz Matrices

Square-root (in particular, Cholesky) factorization of Toeplitz matrices and of their inverses is a classical area of research. The Schur algorithm yields directly the Cholesky factorization of a symmetric Toeplitz matrix, whereas the Levinson algorithm does the same for the inverse matrix. The objective of this letter is to use results from the theory of rational orthonormal functions to derive square-root factorizations of the inverse of an ntimesn positive definite Toeplitz matrix. The main result is a new factorization based on the Takenaka-Malmquist functions, that is parameterized by the roots of the corresponding auto-regressive polynomial of order n . We will also discuss briefly the connection between our analysis and some classical results such as Schur polynomials and the Gohberg-Semencul inversion formula.

Computing a Lower Bound of the Smallest Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix

In this correspondence, several algorithms to compute a lower bound of the smallest eigenvalue of a symmetric positive-definite Toeplitz matrix are described and compared in terms of accuracy and computational efficiency. Exploiting the Toeplitz structure of the considered matrix, new theoretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided.

A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the smallest eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix is derived. The algorithm relies on the computation of the R factor of the QR-factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.

Computation of the Newton step for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix

We compute the Newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix as the reciprocal of the trace of an appropriate matrix. We show that, after the Yule–Walker equations are solved, this trace can be computed in O ( n ) {\mathcal O}(n) additional arithmetic operations, which is in contrast to existing methods, which rely on a recursion, requiring O ( n 2 ) {\mathcal O}(n^2) additional arithmetic operations.

Recurrence relations for the even and odd characteristic polynomials of a symmetric Toeplitz matrix

In a recent paper (Computation of the smallest even and odd eigenvalues of a symmetric positive-definite Toeplitz matrix, SIAM J. Matrix Anal. Appl. 25 (2004) 949–963) Melman proved a recurrence relation of the even and odd characteristic polynomials of a real symmetric Toeplitz matrix T on which a symmetry exploiting method for computing the smallest eigenvalue of T can be based. In this note, we present a proof of the recurrence relation which is less technical and more transparent.

Computation of the Smallest Even and Odd Eigenvalues of a Symmetric Positive-Definite Toeplitz Matrix

We propose an algorithm to compute the smallest even and odd eigenvalues of a real symmetric positive-definite Toeplitz matrix, which is based on the factorization of the characteristic polynomial into an even and an odd polynomial. Newton's method is used to compute the smallest even and odd eigenvalues as the smallest roots of the even and odd characteristic polynomials, respectively.

Detection-estimation of more uncorrelated gaussian sources than sensors in nonuniform linear antenna arrays-part II: partially augmentable arrays

Previous studies dealing with direction-of-arrival (DOA) estimation for uncorrelated planes waves incident on nonuniform M-sensor arrays assumed that the number of signal sources m was known or had already been estimated. In the case (m<M), traditional detection techniques such as Akaike's information criterion (AIC) and minimum description length (MDL) that are based on the equality of several smallest eigenvalues in the covariance matrix may be applied, although we demonstrate that these results can be misleading for nonuniform arrays. In the case (m/spl ges/M), these standard techniques are not applicable. We introduce a new approach to the detection problem for fully augmentable arrays (whose set of intersensor differences is complete). We show that the well-known direct augmentation approach applied to the sample covariance matrix is not a solution by itself since the resulting Toeplitz matrix is generally not positive definite for realistic sample volumes. We propose a transformation of this augmented matrix into a positive definite Toeplitz matrix T/sub /spl mu// with the proper number of equal minimum eigenvalues that are appropriate for the candidate number of sources /spl mu/. Comparison of the results of these best-fit transformations over the permissible range of candidates then allows us to select the most likely number of sources m/spl circ/ using traditional criteria and yields uniquely defined DOAs. Simulation results demonstrate the high performance of this method. Since detection techniques for superior scenarios have not been previously described in the literature, we compare our method with the standard AIC and MDL techniques in a conventional case with similar Cramer-Rao bound (CRB) and find that it has a similar detection performance.

Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method

This paper studies the problem of finding the nearest symmetric positive definite Toeplitz matrix to a given symmetric one. Additional design constraints, which are also formed as closed convex sets in the real Hilbert space of all symmetric matrices, are imposed on the desired matrix. An algorithmic solution to the problem given by the hybrid steepest descent method is established also in the case of inconsistent design constraints.

Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays .I. Fully augmentable arrays

Previous studies dealing with direction-of-arrival (DOA) estimation for uncorrelated planes waves incident on nonuniform M-sensor arrays assumed that the number of signal sources m was known or had already been estimated. In the "conventional" case (m<M), traditional detection techniques such as Akaike's information criterion (AIC) and minimum description length (MDL) that are based on the equality of several smallest eigenvalues in the covariance matrix may be applied, although we demonstrate that these results can be misleading for nonuniform arrays. In the "superior" case (m/spl ges/M), these standard techniques are not applicable. We introduce a new approach to the detection problem for "fully augmentable" arrays (whose set of intersensor differences is complete). We show that the well-known direct augmentation approach applied to the sample covariance matrix is not a solution by itself since the resulting Toeplitz matrix is generally not positive definite for realistic sample volumes. We propose a transformation of this augmented matrix into a positive definite Toeplitz matrix T/sub /spl mu// with the proper number of equal minimum eigenvalues that are appropriate for the candidate number of sources /spl mu/. Comparison of the results of these best-fit transformations over the permissible range of candidates then allows us to select the most likely number of sources m/spl circ/ using traditional criteria and yields uniquely defined DOAs. Simulation results demonstrate the high performance of this method. Since detection techniques for superior scenarios have not been previously described in the literature, we compare our method with the standard AIC and MDL techniques in a conventional case with similar Cramer-Rao bound (CRB) and find that it has a similar detection performance.

A symmetry exploiting Lanczos method for symmetric Toeplitz matrices

Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. Most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach.

Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.