scholarly journals On decomposition of k-tridiagonal ℓ-Toeplitz matrices and its applications

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. Ohashi ◽  
T. Sogabe ◽  
T.S. Usuda
Keyword(s):  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Arbitrary Integer ◽  
The Matrix ◽  
Small Matrix

AbstractWe consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.

scholarly journals Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences

2021 ◽  
Vol 37 ◽  
pp. 370-386
Author(s):  
Paola Ferrari ◽  
Isabella Furci ◽  
Stefano Serra-Capizzano
Keyword(s):  
Toeplitz Matrix ◽  
Integrable Function ◽  
Toeplitz Matrices ◽  
Singular Value ◽  
Value Distribution ◽  
Identity Matrix ◽  
Matrix Size ◽  
Lebesgue Integrable Function ◽  
Matrix Sequence ◽  
The Matrix

In recent years,  motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, the authors consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

scholarly journals REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

2019 ◽  
Vol 7 ◽  
Author(s):  
ANIRBAN BASAK ◽  
ELLIOT PAQUETTE ◽  
OFER ZEITOUNI
Keyword(s):  
Toeplitz Matrices ◽  
Random Variable ◽  
Singular Values ◽  
Equivalence Theorem ◽  
Empirical Measure ◽  
Uniform Random Variable ◽  
Normal Matrices ◽  
Nilpotent Matrix ◽  
The Matrix ◽  
Triangular Toeplitz Matrix

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let$M_{N}$be a deterministic$N\times N$matrix, and let$G_{N}$be a complex Ginibre matrix. We consider the matrix${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where$\unicode[STIX]{x1D6FE}>1/2$. With$L_{N}$the empirical measure of eigenvalues of${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties$L_{N}$to the singular values of$z-M_{N}$, with$z\in \mathbb{C}$. We then compute the limit of$L_{N}$when$M_{N}$is an upper-triangular Toeplitz matrix of finite symbol: if$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$where$\mathfrak{d}$is fixed,$a_{i}\in \mathbb{C}$are deterministic scalars and$J$is the nilpotent matrix$J(i,j)=\mathbf{1}_{j=i+1}$, then$L_{N}$converges, as$N\rightarrow \infty$, to the law of$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$where$U$is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when$\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in$M_{N}$.

On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices

2018 ◽  
Vol 33 ◽  
pp. 74-82 ◽  
Author(s):  
Katarzyna Filipiak ◽  
Augustyn Markiewicz ◽  
Adam Mieldzioc ◽  
Aneta Sawikowska
Keyword(s):  
Toeplitz Matrices ◽  
Positive Definite Matrix ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Asymptotic Cone ◽  
The Matrix ◽  
Multi Level ◽  
Nonnegative Definite ◽  
Unknown Covariance ◽  
Numer Math

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

scholarly journals Statistical applications for equivariant matrices

2001 ◽  
Vol 25 (1) ◽  
pp. 53-61
Author(s):  
S. H. Alkarni
Keyword(s):  
Linear Systems ◽  
Fourier Transforms ◽  
Toeplitz Matrices ◽  
Computational Cost ◽  
Special Kind ◽  
Circulant Matrices ◽  
Stationary Processes ◽  
Linear System Of Equations ◽  
The Matrix ◽  
A Finite Group

Solving linear system of equationsAx=benters into many scientific applications. In this paper, we consider a special kind of linear systems, the matrixAis an equivariant matrix with respect to a finite group of permutations. Examples of this kind are special Toeplitz matrices, circulant matrices, and others. The equivariance property ofAmay be used to reduce the cost of computation for solving linear systems. We will show that the quadratic form is invariant with respect to a permutation matrix. This helps to know the multiplicity of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. Applications for such systems from the area of statistics will be presented. These include Fourier transforms on a symmetric group as part of statistical analysis of rankings in an election, spectral analysis in stationary processes, prediction of stationary processes and Yule-Walker equations and parameter estimation for autoregressive processes.

Least squares Toeplitz matrix solutions of the matrix equation

2016 ◽  
Vol 65 (9) ◽  
pp. 1867-1877 ◽  
Author(s):  
Yongxin Yuan ◽  
Wenhua Zhao ◽  
Hao Liu
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Toeplitz Matrix ◽  
The Matrix

Accelerating and tuning small matrix multiplications on Sunway TaihuLight: A case study of spectral element CFD Code Nek5000

2019 ◽  
Vol 34 (2) ◽  
pp. 178-186
Author(s):  
Xianmeng Wang ◽  
Zhifeng Zhou ◽  
Changjun Hu ◽  
Wen Yang ◽  
Minfu Zhao ◽  
...  
Keyword(s):  
Spectral Element ◽  
Dense Matrix ◽  
Heterogeneous Architecture ◽  
Performance Improvements ◽  
Matrix Products ◽  
The Matrix ◽  
Sunway Taihulight ◽  
Small Matrix ◽  
Computational Fluid Dynamics Cfd

The matrix–matrix products for matrices of small size have continued to play an important part in a range of scientific applications. The heterogeneous architecture, which is predicted to be a trend in the exascale supercomputing era, gives rises to the challenges of porting and optimizing small matrix products. We present a method to accelerating and tune small matrix multiplications on Sunway TaihuLight supercomputer, which has been titled as the most powerful supercomputer four times in the Top5000 list. Sunway TaihuLight is equipped with Shen-Wei hybrid manycore processors. We use Nek5000 as a case study to demonstrate our methods. Nek5000 is an open-source computational fluid dynamics (CFD) solver based on the spectral element method (SEM) for incompressible flow. The high-order SEM method, of which the computation kernel is small dense matrix products, is regarded to have the potential to overcome constraints of standard CFD software. By optimizing using vectorization, we gained about 30% performance improvement on management processing element. We accelerated Nek5000 using computing processing elements (CPEs). The experiments results suggest that employing 32 CPEs delivers the best performance enhancements. We scaled Nek5000 to 16,384 core groups with 540,672 cores, reaching about 30% performance improvements.

scholarly journals A note on multilevel Toeplitz matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 114-126
Author(s):  
Lei Cao ◽  
Selcuk Koyuncu
Keyword(s):  
Toeplitz Matrix ◽  
Symmetric Matrix ◽  
Toeplitz Matrices ◽  
Tensor Products ◽  
Symmetric Matrices ◽  
Unitary Matrices ◽  
Complex Symmetric Matrix ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices

Abstract Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

scholarly journals The Inverses of Block Toeplitz Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiao-Guang Lv ◽  
Ting-Zhu Huang
Keyword(s):  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Circulant Matrices ◽  
Scalar Case ◽  
Block Toeplitz Matrices ◽  
Sum Of Products ◽  
Cyclic Displacement

We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.

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