Toeplitz Systems Research Articles

Inverse Toeplitz preconditioners for Hermitian Toeplitz systems

AbstractIn this paper we consider solving Hermitian Toeplitz systems Tnx=b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non‐negative continuous 2π‐periodic function ƒ, i.e. Tn=𝒯n[ƒ]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if ƒ is positive then the spectrum of 𝒯n[1/ƒ]𝒯n[ƒ] is clustered around 1. We prove that the trigonometric polynomial q (s⩾2, cf. (2) and (3)) converges to 1/ƒ uniformly as n→∞ under the condition that 1/ƒ is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/ƒ with q, where N<n. We also extend our method to construct efficient preconditioners for Tn when ƒ has finite zeros of even orders. We prove that with our preconditioners, the preconditioned matrix has spectrum clustered around 1. It follows that the PCG methods converge very fast when applied to solve the preconditioned systems. Numerical results are given to demonstrate the efficiency of our preconditioners. Copyright © 2004 John Wiley & Sons, Ltd.

A family of modified regularizingcirculant preconditioners for two-levels Toeplitz systems

Customary circulant preconditioners of Chan, Strang, and inverse-Toeplitz type aremodified for the regularized solution of ill-conditioned Toeplitz systems. Given a symmetric two-level Toeplitz matrix A, and two parameters τ and σ, consider the customary preconditioner C and modify it by replacing its eigenvalues with modulus less than τ with the value σ. The circulant matrix obtained in this way is used as a preconditioner. We analyze the distribution of the eigenvalues of the preconditioned matrix and the regularization properties. Numerical experiments, shown for the image restoration problem, validate the theoretical results.

Practical Band Toeplitz Preconditioning and Boundary Layer Effects

In the last decade many efficient iterative solvers for n×n Hermitian positive definite Toeplitz systems have been devised. Many of them are based on band Toeplitz preconditioners: they are optimal but require the knowledge of the zeros of the underlying generating function. In some cases this information is available and in some cases is not. In [27] an economic numerical procedure for finding these zeros within a given precision has been devised. Here we provide conditions on the approximation error of these zeros in order to maintain the optimality that is a convergence rate independent of the dimension n of the considered linear systems.

W Circulant Boundary Condition

AbstractWe use the preconditioned conjugate gradient (PCG) method to analyze the symmetric positive‐definite Toeplitz systems. Then we present a new embedding constructing preconditioner way and prove that many other ways of constructing preconditioners are generally the special cases of this method. We propose the w‐circulant boundary condition and make comparison with both the ordinary circulant boundary condition and helix one. Furthermore, we obtain a new type of boundary condition: the mixed boundary condition.

The solution of ill-conditioned symmetric toeplitz systems via two-grid and wavelet methods

In this paper, we present a generalized multigrid method combined with wavelet filters for solving ill-conditioned symmetric Toeplitz systems T n x = b, where T n ϵ R n× n is generated by nonnegative functions with zeros. First, we propose the construction of general Cohen, Daubechies, and Feauveau (CDF) 9/7 biorthogonal wavelet systems, so that a new class of compactly supported biorthogonal wavelet systems GCDF are achieved with specified vanishing moments for scaling functions. In order to solve ill-conditioned Toeplitz systems by using the two-grid method (TGM), we use the constructed GCDF wavelets to get prolongation and restriction operators. As a result, the proposed TGM is proved by numerical experiment to be more efficient than the classic TGM, especially when T n is seriously ill-conditioned. For the vanishing moments being N, experimental tests illustrate that the TGM with damped-Jacobi smoother converges when the generating function has zeros of order less than or equal to 2N ( N ≤ 8). Besides, we prove in theory that the proposed method converges for Toeplitz systems that are generated by functions with zeros of order less than or equal to four.

Circulant and skew-circulant splitting methods for Toeplitz systems

We study efficient iterative methods for Toeplitz systems based on the circulant and skew-circulant splitting (CSCS) of the Toeplitz matrix. Theoretical analysis show that if the circulant and the skew-circulant splitting matrices are positive definite, then the CSCS method converges to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew-circulant matrices involved. Numerical examples are presented to demonstrate the method.

Some algorithms for solving special tridiagonal block Toeplitz linear systems

This paper is focused on different methods and algorithms for solving tridiagonal block Toeplitz systems of linear equations. We consider the El-Sayed method (Ph.D. Thesis, 1996) for such systems and propose several modifications that lead to different algorithms, which we discuss in detail. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Comments and conclusions for computing efficiency of the proposed new algorithms are given. Numerical experiments corroborating the theoretical results are also presented.

How to handle colored observation noise in large least-squares problems

An approach to handling colored observation noise in large least-squares (LS) problems is presented. The handling of colored noise is reduced to the problem of solving a Toeplitz system of linear equations. The colored noise is represented as an auto regressive moving-average (ARMA) process. Stability and invertability of the ARMA model allows the solution of the Toeplitz system to be reduced to two successive filtering operations using the inverse transfer function of the ARMA model. The numerical complexity of the algorithm scales proportionally to the order of the ARMA model times the number of observations. This makes the algorithm particularly suited for LS problems with millions of observations. It can be used in combination with direct and iterative algorithms for the solution of the normal equations. The performance of the algorithm is demonstrated for the computation of a model of the Earth’s gravity field from simulated satellite-derived gravity gradients up to spherical harmonic degree 300.

Centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices and Bezoutians

In the paper algebraic properties of centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel (T+H) matrices are investigated. It is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related Toeplitz systems with symmetric or skewsymmetric right-hand sides. In the singular case the related Toeplitz matrices determine the kernel of the T+H matrices. Finally centrosymmetric and centro-skewsymmetric T+H Bezoutians are characterized.

Semi-infinite multi-index perturbed block Toeplitz systems

In this article Banach algebra techniques are employed to study the numerical solution of linear systems with a semi-infinite multi-index suitably perturbed k× k block Toeplitz matrix. Decay properties of their solutions are studied by using suitably weighted Wiener algebras. A projection-type method for their numerical solution is introduced. Numerical results are presented illustrating both the accuracy of the method for certain perturbed Toeplitz systems and the need for generalizing the theoretical framework to other such systems.

Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions

In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach.

A Superfast Toeplitz Solver with Improved Numerical Stability

This paper describes a new O(n log3(n)) solver for the positive definite Toeplitz system Tx=b. Instead of computing generators for the inverse of T, the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The generators of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well-known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases.

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Summary. The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13, 14, 22]. The technique was modified in [6, 36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at x 0 =0 of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at x 0 =0 of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed.

The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case

In [10,14], circulant-type preconditioners have been proposed for ill-conditioned Hermitian Toeplitz systems that are generated by nonnegative continuous functions with a zero of even order. The proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero. Therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems. In [10,14], it was claimed that this result can be the case where the generating functions have multiple zeros. The main aim of this paper is to give a complete convergence proof of the method in [10,14] for this class of generating functions.

A parallel divide and conquer algorithm for non symmetric tridiAgonal toeplitz systems using conjugate gradient

Abstract In this paper, we consider the application of the conjugate gradient method specifically to solve non symmetric systems which are large, tridiagonal and Toeplitz. Under the condition that the system is diagonally dominant, one can pre-multiply the system by the transpose of the coefficient matrix and take advantage of the structure of the new coefficient matrix to perturb and factor it. This allows us to divide the task of solution containing pairs of tridiagonal, symmetric and Toeplitz systems and to solve the pairs of systems using a parallel implementaton of congujate gradient. Final corrections, to account for the perturbations, provide a numerical approximation to the solution.

Four short stories about Toeplitz matrix calculations

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Fast direct solution methods for symmetric banded Toeplitz systems, based on the sine transform

We present new fast direct methods for solving a large symmetric banded Toeplitz system of order n with bandwidth p. We make use of structured matrices which can be diagonalized by the discrete sine transform matrix, sometimes called τ-matrices. A first method writes the Toeplitz matrix as the sum of a τ-matrix and a low rank matrix. A second method embeds the Toeplitz matrix in a larger τ-matrix of order m. The methods are similar to Jain [IEEE Trans. Acoust. Speech Signal Process. 26 (1978) 121] and Linzer [Linear Algebra Appl. 170 (1992) 1], who worked with circulant matrices. Both algorithms consist in solving two τ-systems and two smaller systems. A τ-system of order n can be solved in O( nlog n) by using a discrete sine transform if n+1 has small prime factors. Therefore, the second algorithm is preferable, since we can choose m such that m+1 has small prime factors. On the other hand, in the second method the smaller systems can become large when m differs too much from n, while in the first method the order is always p−1. In both methods, the small systems have low displacement rank, so we can use fast methods to solve them.

Computations with infinite Toeplitz matrices and polynomials

We relate polynomial computations with operations involving infinite band Toeplitz matrices and show applications to the numerical solution of Markov chains, of nonlinear matrix equations, to spectral factorizations and to the solution of finite Toeplitz systems. In particular two matrix versions of Graeffe's iteration are introduced and their convergence properties are analyzed. Correlations between Graeffe's iteration for matrix polynomials and cyclic reduction for block Toeplitz matrices are pointed out. The paper contains a systematic treatment of known topics and presentation of new results, improvements and extensions.

A note on the fast algorithm for block Toeplitz systems with tensor structure

We study the solutions of block Toeplitz systems T mn x= b by using the preconditioned conjugate gradient (PCG) method. Here T mn = T m ⊗ T n and T i, i=m,n are Toeplitz matrices. In [X. Jin, Appl. Math. Comput. 73 (1995) 115–124], Jin introduced a fast algorithm for these systems by applying the PCG method. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn-by- mn system is well conditioned, then the PCG method converges superlinearly and only O( mnlog mn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in Jin, 1995. In this paper, we apply this fast algorithm with the { ω}-circulant preconditioners proposed in [D. Potts, G. Steidl, Preconditioners for Ill-Conditioned toeplitz matrices, BIT, to be appeared] to solve the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of the algorithm for solving the preconditioned systems by using the PCG method. An application in image restoration is also given.

A PARALLEL DIVIDE AND CONQUER ALGORITHM FOR NON SYMMETRIC TRIDIAGONAL TOEPLITZ SYSTEMS USING CONJUGATE GRADIENT

In this paper, we consider the application of the conjugate gradient method specifically to solve non symmetric systems which are large, tridiagonal and Toeplitz. Under the condition that the system is diagonally dominant, one can pre-multiply the system by the transpose of the coefficient matrix and take advantage of the structure of the new coefficient matrix to perturb and factor it. This allows us to divide the task of solution containing pairs of tridiagonal, symmetric and Toeplitz systems and to solve the pairs of systems using a parallel implementaton of congujate gradient. Final corrections, to account for the perturbations, provide a numerical approximation to the solution.