Toeplitz matrix

About: Toeplitz matrix is a research topic. Over the lifetime, 8097 publications have been published within this topic receiving 131901 citations.

Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix

Abstract: Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid $\Omega$. This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over $m+1$ equispaced points on a line can be produced at the cost of an initial FFT of length $2m$ with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an $(m+1)\times(m+1) $ Toeplitz correlation matrix $R$ can be embedded in a nonnegative definite $2M\times2M$ circulant matrix $S$, exact realizations of the normal multivariate $y \sim {\cal N}(0,R)$ can be generated via FFTs of length $2M$. Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which $M = m$ is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated.

An Algorithm for the Inversion of Finite Toeplitz Matrices

Abstract: possesses properties (1.1), (1.2), and (1.3). In order to find the joint probability density function of (yo, yi , , yn), or of any n + 1 successive variates, it is necessary to invert the matrix Tn . In this paper, we derive an exact recursive procedure for the numerical inversion of an arbitrary positive definite Toeplitz matrix of finite order, which takes full advantage of the strong restrictions placed on its elements by (1.1), (1.2), and (1.3). The number of multiplications required for the inversion of an nth order Toeplitz matrix, using this procedure, is proportional to n2, rather than to n', as in the case of methods which are suitable for arbitrary Hermitian matrices. To the author's knowledge, this inversion algorithm is the first to be specifically designed to take advantage of the peculiar simplicity of the general Toeplitz matrix. In addition, the closing section of the paper is devoted to a statement of an algorithm for the inversion of lnon-Hernmitian matrices of the form (1.1).

Operators, Functions, and Systems: An Easy Reading

Abstract: Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS, 2002, this unique work combines four major topics of modern analysis and its applications: A. Hardy classes of holomorphic functions, B. Spectral theory of Hankel and Toeplitz operators, C. Function models for linear operators and free interpolations, and D. Infinite-dimensional system theory and signal processing. This volume contains Parts A and B. Hardy classes of holomorphic functions is known to be the most powerful tool in complex analysis for a variety of applications, starting with Fourier series, through the Riemann $\zeta$-function, all the way to Wiener's theory of signal processing. Spectral theory of Hankel and Toeplitz operators becomes the supporting pillar for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, Nevanlinna-Pick and Carathodory interpolation, and the best rational approximations are considered to illustrate the power of Hankel and Toeplitz operators. The book is geared toward a wide audience of readers, from graduate students to professional mathematicians, interested in operator theory and functions of a complex variable. The two volumes develop an elementary approach while retaining an expert level that can be applied in advanced analysis and selected applications. Readership Graduate students and research mathematicians interested in analysis.

Lattice filters for adaptive processing

Abstract: This paper presents a tutorial review of lattice structures and their use for adaptive prediction of time series Lattice filters associated with stationary covariance sequences and their properties are discussed The least squares prediction problem is defined for the given data case, and it is shown that many of the currently used lattice methods are actually approximations to the stationary least squares solution The recently developed class of adaptive least squares lattice algorithms are described in detail, both in their unnormalized and normalized forms The performance of the adaptive least squares lattice algorithm is compared to that of some gradient adaptive methods Lattice forms for ARMA processes, for joint process estimation, and for the sliding-window covariance case are presented The use of lattice structures for efficient factorization of covariance matrices and solution of Toeplitz sets of equations is briefly discussed

An Optimal Circulant Preconditioner for Toeplitz Systems

Abstract: Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\|C - A\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.