Symmetric Positive Definite Toeplitz Matrix Research Articles

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.

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.

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.

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.

Computing the Smallest Eigenpair of a Symmetric Positive Definite Toeplitz Matrix

An algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix is presented. The eigenvalue is approximated from below by Newton's method applied to the characteristic polynomial of the matrix. The Newton's step is calculated by a Levinson--Durbin type recursion. Simultaneously, this recursion produces a realistic error bound of the actual approximation without additional computing effort as well as a simple and efficient way to compute the associated eigenvector.

Computing the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix by Newton-type Methods

A method for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix by application of Newton's method to the characteristic polynomial has been recently introduced by Mastronardi and Boley [ SIAM J. Sci. Comput., 20 (1999), pp. 1921--1927]. Though considerably slower than methods developed by the authors [W. Mackens and H. Voss, SIAM J. Matrix. Anal. Appl., 18 (1997), pp. 521--534], [W. Mackens and H. Voss, Linear Algebra Appl., 275/276 (1998), pp. 401--415], [H. Voss, Linear Algebra Appl., 287 (1999), pp. 359--371] the new approach is conceptually much simpler. In this paper we improve the performance of the new method substantially while keeping its simplicity.

A projection method for computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix

A projection method for computing the minimal eigenvalue of a symmetric and positive definite Toeplitz matrix is presented. It generalizes and accelerates the algorithm considered in [12] (W. Mackens, H. Voss, SIAM J. Matrix Anal. Appl. 18 (1997) 521–534). Global and cubic convergence is proved. Randomly generated test problems up to dimension 1024 demonstrate the methods good global behaviour.

The Minimum Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix and Rational Hermitian Interpolation

A novel method for computing the minimal eigenvalue of a symmetric positive-definite Toeplitz matrix is presented. Similar to the algorithm of Cybenko and Van Loan, it is a combination of bisection and a root finding method. Both phases of the method are accelerated considerably by rational Hermitian interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which must be solved to determine the smallest eigenvalue is 6.6, which reduces the computational cost of the method of Cybenko and Van Loan to approximately 35%. The method includes a rigorous error bound.

On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms

This report contains a numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2. The algorithms in the class can be expressed as sequences of elementary downdating steps. The stability of the factorization algorithms follows directly from the numerical properties of algorithms for realizing elementary downdating operations. It is shown that the Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix is in the class and hence the Bareiss algorithm is stable. Some numerical experiments that compare behavior of the Bareiss algorithm and the Levinson algorithm are presented. These experiments indicate that in general (when the reflection coefficients are not all positive) the Levinson algorithm is not stable; certainly it can give much larger residuals than the Bareiss algorithm.

MinimaL eigenvalue of a real symmetric positive definite Toeplitz matrix

The problem of computing the minimal eigenvalue of a real symmetric positive definite Toeplitz matrix is considered. Algorithms for estimating such an eigenvalue, which need only one or two inverses, are presented. The suggested algorithms are based on good initial approximations to the corresponding eigenvector which can be derived by approximating the Toeplitz matrix by circulant matrices.

On the Inverse M-Matrix Problem for Real Symmetric Positive-Definite Toeplitz Matrices

Necessary and sufficient conditions are obtained for a real symmetric positive-definite Toeplitz matrix R to be an inverse of an M-matrix in terms of its Schur coefficients. Related problems are also considered, such as when such a matrix R can be extended to a higher-dimensional real symmetric positive-definite Toeplitz matrix whose inverse is an M-matrix or, under less restrictive conditions on R, when only its Cholesky factors are inverses of M-matrices. The proofs are constructive and allow the generation of such R’s with the various aforementioned properties.

Computing the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

A method for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix is given. It relies solely upon the Levinson–Durbin algorithm. The procedure involves a combination of bisection and Newton's method. Good starting values are also shown to be obtainable from the Levinson–Durbin algorithm.

On the infinity norm of the inverse of a class of toeplitz matrices of band width five

A useful theorem for precisely bounding the infinity norm of the inverse of a five-diagonal symmetric positive definite Toeplitz matrix is obtained.

Properties of the inverses of a set of band matrices

Those regions where the elements of the inverse of a Toeplitz matrix of band width four and order nalternate in sign are determined. A similar result concerning the elements of the inverse of a commonly occurring symmetric positive definite Toeplitz matrix of band width five and order nis extended to cover the case when the first and last rows of the matrix are modified through a change in the boundary conditions of the associated application. A method for economically obtaining the infinity norm of the pent-diagonal symmetric positive definite Toeplitz matrix is then derived.

A derivation of the Levinson algorithm for solving linear systems with symmetric positive definite Toeplitz matrix

Based on an orthogonalization technique, published earlier in this journal, a derivation is given of the Levinson algorithm for solving systems with a symmetric positive definite Toeplitz matrix.