Optimal preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make them converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be optimal. In contrast, for multilevel Toeplitz matrices, there...

Abstract. 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 solu...

In this paper, we consider robust system identification under sparse outliers and random noises. In this problem, system parameters are observed through a Toeplitz matrix. All observations are subject to random noises and a few are corrupted with outliers. We reduce this problem of system identification to a sparse error correcting problem using a Toeplitz structured real-numbered coding matrix...

When the Inverse Additive Singular Value Problem (IASVP) involves Toeplitz–type matrices it is possible to exploit this special structure to reduce the execution time. In this paper, we present two iterative local and global convergent algorithms (MIIIT and LPT) to solve efficiently the IASVP when the matrix is Toeplitz (IASVPT). As it will be shown, it can be achieved an asymptotic complexity ...

The solution of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang and analyzed by R. Chan and Strang. The convergence rate of the PCG method depends heavily on the choice of preconditioners for the given Toeplitz matrices. In this paper, we present a general approach to the design of Toeplitz preconditioners...

We apply the superfast divide-and-conquer MBA algorithm to possibly singular n × n Toeplitz-like integer input matrices and extend it to computations in the ring of integers modulo a power of a random prime. We choose the power which barely fits the size of a computer word; this saves word operations in the subsequent lifting steps. We extend our early techniques for avoiding degeneration while...

An exact formula was recently obtained for the spectral norms of the Lucas and Fibonacci Hankel matrices [1], and also for the Lucas and Fibonacci Toeplitz matrices [4]. These results put finishing touches on the works initiated in [2] and [3]. On another front, bounds were found for the spectral norms of k-Fibonacci and k-Lucas Toeplitz matrices [7]. In this paper, we present the exact value f...

The authors use results from [6, 7] to analyze the asymptotics of eigenvalues of Toeplitz matrices with certain continuous and discontinuous symbols. In particular, the authors prove a conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitz eigenvalues.

The concept of the hafnian first appeared in works on
 quantum field theory by E. R. Caianiello. However, it also has
 an important combinatorial property: adjacency
 matrix undirected weighted graph is equal to the
 total sum weights perfect matchings this graph.
 In general, use limited complexity
 its computation. paper, we present a method for exact calculation...

In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (ω)circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz matrices, especially in the real case...