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10 pages, 253 KiB

Open AccessArticle

Nonnegative Inverse Elementary Divisors Problem for Lists with Nonnegative Real Parts

by Hans Nina, Hector Flores Callisaya, H. Pickmann-Soto and Jonnathan Rodriguez

Mathematics 2020, 8(10), 1662; https://doi.org/10.3390/math8101662 - 27 Sep 2020

Viewed by 1635

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb [...] Read more.

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

13 pages, 304 KiB

Open AccessArticle

Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

by Martin Gavalec, Ján Plavka and Daniela Ponce

Mathematics 2020, 8(8), 1348; https://doi.org/10.3390/math8081348 - 12 Aug 2020

Cited by 5 | Viewed by 2153

Abstract

The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, [...] Read more.

The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, or activations of some other cooperating machines, is described by a systems of max-plus linear equations. In particular, if the system is in a steady state, such as a synchronized computer network in data processing, then the state vector is an eigenvector of the system. In reality, the entries of matrices and vectors are considered as intervals. The properties and recognition algorithms for several types of interval eigenvectors are studied in this paper. For a given interval matrix and interval vector, a set of generators is defined. Then, the strong and the strongly universal eigenvectors are studied and described as max-plus linear combinations of generators. Moreover, a polynomial recognition algorithm is suggested and its correctness is proved. Similar results are presented for the weak eigenvectors. The results are illustrated by numerical examples. The results have a general character and can be applied in every max-plus algebra and every instance of the interval eigenproblem. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

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14 pages, 345 KiB

Open AccessArticle

Tensor Global Extrapolation Methods Using the n-Mode and the Einstein Products

by Alaa El Ichi, Khalide Jbilou and Rachid Sadaka

Mathematics 2020, 8(8), 1298; https://doi.org/10.3390/math8081298 - 5 Aug 2020

Cited by 9 | Viewed by 3180

Abstract

In this paper, we present new Tensor extrapolation methods as generalizations of well known vector, matrix and block extrapolation methods such as polynomial extrapolation methods or

ϵ

-type algorithms. We will define new tensor products that will be used to introduce global tensor [...] Read more.

In this paper, we present new Tensor extrapolation methods as generalizations of well known vector, matrix and block extrapolation methods such as polynomial extrapolation methods or

ϵ

-type algorithms. We will define new tensor products that will be used to introduce global tensor extrapolation methods. We discuss the application of these methods to the solution of linear and non linear tensor systems of equations and propose an efficient implementation of these methods via the global-QR decomposition. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

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12 pages, 279 KiB

Open AccessArticle

Characteristic Polynomials and Eigenvalues for Certain Classes of Pentadiagonal Matrices

by María Alejandra Alvarez, André Ebling Brondani, Francisca Andrea Macedo França and Luis A. Medina C.

Mathematics 2020, 8(7), 1056; https://doi.org/10.3390/math8071056 - 1 Jul 2020

Cited by 3 | Viewed by 3008

Abstract

There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic [...] Read more.

There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomial and explicit formulas for their eigenvalues. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

15 pages, 361 KiB

Open AccessArticle

Applications of the Periodogram Method for Perturbed Block Toeplitz Matrices in Statistical Signal Processing

by Jesús Gutiérrez-Gutiérrez, Xabier Insausti and Marta Zárraga-Rodríguez

Mathematics 2020, 8(4), 582; https://doi.org/10.3390/math8040582 - 14 Apr 2020

Cited by 1 | Viewed by 1909

Abstract

In this paper, we combine the periodogram method for perturbed block Toeplitz matrices with the Cholesky decomposition to give a parameter estimation method for any perturbed vector autoregressive (VAR) or vector moving average (VMA) process, when we only know a perturbed version of [...] Read more.

In this paper, we combine the periodogram method for perturbed block Toeplitz matrices with the Cholesky decomposition to give a parameter estimation method for any perturbed vector autoregressive (VAR) or vector moving average (VMA) process, when we only know a perturbed version of the sequence of correlation matrices of the process. In order to combine the periodogram method for perturbed block Toeplitz matrices with the Cholesky decomposition, we first need to generalize a known result on the Cholesky decomposition of Toeplitz matrices to perturbed block Toeplitz matrices. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

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12 pages, 243 KiB

Open AccessArticle

A Note on NIEP for Leslie and Doubly Leslie Matrices

by Luis Medina, Hans Nina and Elvis Valero

Mathematics 2020, 8(4), 559; https://doi.org/10.3390/math8040559 - 10 Apr 2020

Cited by 2 | Viewed by 2319

Abstract

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is [...] Read more.

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

21 pages, 568 KiB

Open AccessArticle

Generalized Structure Preserving Preconditioners for Frame-Based Image Deblurring

by Davide Bianchi and Alessandro Buccini

Mathematics 2020, 8(4), 468; https://doi.org/10.3390/math8040468 - 27 Mar 2020

Cited by 4 | Viewed by 2275

Abstract

We are interested in fast and stable iterative regularization methods for image deblurring problems with space invariant blur. The associated coefficient matrix has a Block Toeplitz Toeplitz Blocks (BTTB) like structure plus a small rank correction depending on the boundary conditions imposed on [...] Read more.

We are interested in fast and stable iterative regularization methods for image deblurring problems with space invariant blur. The associated coefficient matrix has a Block Toeplitz Toeplitz Blocks (BTTB) like structure plus a small rank correction depending on the boundary conditions imposed on the imaging model. In the literature, several strategies have been proposed in the attempt to define proper preconditioner for iterative regularization methods that involve such linear systems. Usually, the preconditioner is chosen to be a Block Circulant with Circulant Blocks (BCCB) matrix because it can efficiently exploit Fast Fourier Transform (FFT) for any computation, including the (pseudo-)inversion. Nevertheless, for ill-conditioned problems, it is well known that BCCB preconditioners cannot provide a strong clustering of the eigenvalues. Moreover, in order to get an effective preconditioner, it is crucial to preserve the structure of the coefficient matrix. On the other hand, thresholding iterative methods have been recently successfully applied to image deblurring problems, exploiting the sparsity of the image in a proper wavelet domain. Motivated by the results of recent papers, the main novelty of this work is combining nonstationary structure preserving preconditioners with general regularizing operators which hold in their kernel the key features of the true solution that we wish to preserve. Several numerical experiments shows the performances of our methods in terms of quality of the restorations. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

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17 pages, 402 KiB

Open AccessArticle

Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

by Paola Ferrari, Ryma Imene Rahla, Cristina Tablino-Possio, Skander Belhaj and Stefano Serra-Capizzano

Mathematics 2020, 8(1), 5; https://doi.org/10.3390/math8010005 - 18 Dec 2019

Cited by 4 | Viewed by 2648

Abstract

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the

Q_{k}

Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is [...] Read more.

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the

Q_{k}

Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is

div (- a (x) \nabla \cdot)

, with a continuous and positive over

\bar{Ω}

,

Ω

being an open and bounded subset of

R^{2}

. While the analysis is performed in one dimension, the numerics are carried out also in higher dimension

d \geq 2

, showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k. Full article

(This article belongs to the Special Issue Matrix Structures: Numerical Methods and Applications)

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