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Mathematics
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Numerical Analysis and Matrix Computations: Theory and Applications
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Numerical Analysis and Matrix Computations: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (15 January 2025) | Viewed by 8795

Special Issue Editor

Prof. Dr. Petko Petkov

E-Mail Website
Guest Editor
Department of Engineering Sciences, Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria
Interests: matrix computations; control theory; robust control; numerical methods; software for matrix computations and control systems design

Special Issue Information

Dear Colleagues,

The area of numerical matrix computations is a field of intensive research and important applications. Matrix computations are used in almost all sciences and engineering including disciplines such as quantum information, mathematical biology, seismology, data science, dynamical systems, control theory, signal and image processing,  geometrical modeling, network theory, and many others.

The journal Mathematics plans to publish a Special Issue devoted to numerical analysis and matrix computations. Prospective authors from different fields of science are invited to share their state-of-the-art research results, present application experience, exchange new ideas, and discuss future developments in the area of numerical matrix computations.

The list of topics covered by the Special Issue includes but is not restricted to the following subjects:

  1. Perturbation analysis of matrix problems;
  2. Ill-conditioned matrix problems and their regularization;
  3. Matrices depending on parameters;
  4. Eigenvalue and eigenspaces sensitivity;
  5. Computations related to matrix pencils and matrix polynomials;
  6. Eigenvalue computations;
  7. Jordan and Schur algorithms in matrix computations;
  8. Numerical solution of matrix equations and linear matrix inequalities;
  9. Computing matrix functions;
  10. Numerical matrix methods for solving difference and differential equations;
  11. Software for matrix computations;
  12. Numerical computations in engineering problems, etc.

Prof. Dr. Petko Petkov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Matrix computations
  • Matrix decomposition
  • Matrix problems
  • Eigenvalue problems
  • Matrix factorization
  • Singular value decomposition
  • Riccati equation
  • Dynamic mode decomposition
  • Linear quadratic Gaussian control
  • Numerical linear algebra
  • Numerical analysis
  • Applications in engineering, mechanics, biology, signal processing, etc.

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (6 papers)

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Research

27 pages, 1332 KiB  
Article
Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures
by Danyal Ahmad, Muhammad Faisal Khan and Stefano Serra-Capizzano
Mathematics 2025, 13(3), 393; https://doi.org/10.3390/math13030393 - 24 Jan 2025
Viewed by 653
Abstract
In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) ∗-algebra. We consider the [...] Read more.
In the current work, we analyze the spectral distribution of the geometric mean of two or more matrix-sequences constituted by Hermitian positive definite matrices, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) ∗-algebra. We consider the geometric mean for two matrices, using the Ando-Li-Mathias (ALM) definition, and then we pass to the extension of the idea to more than two matrices by introducing the Karcher mean. While there is no simple formula for the geometric mean of more than two matrices, iterative methods from the literature are employed to compute it. The main novelty of the work is the extension of the study in the distributional sense when input matrix-sequences belong to one of the GLT ∗-algebras. More precisely, we show that the geometric mean of more than two positive definite GLT matrix-sequences forms a new GLT matrix-sequence, with the GLT symbol given by the geometric mean of the individual symbols. Numerical experiments are reported concerning scalar and block GLT matrix-sequences in both one-dimensional and two-dimensional cases. A section with conclusions and open problems ends the current work. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
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10 pages, 390 KiB  
Article
The High Relative Accuracy of Computations with Laplacian Matrices
by Héctor Orera and Juan Manuel Peña
Mathematics 2024, 12(22), 3491; https://doi.org/10.3390/math12223491 - 8 Nov 2024
Viewed by 648
Abstract
This paper provides an efficient method to compute an LDU decomposition of the Laplacian matrix of a connected graph with high relative accuracy. Several applications of this method are presented. In particular, it can be applied to efficiently compute the eigenvalues [...] Read more.
This paper provides an efficient method to compute an LDU decomposition of the Laplacian matrix of a connected graph with high relative accuracy. Several applications of this method are presented. In particular, it can be applied to efficiently compute the eigenvalues of the mentioned Laplacian matrix. Moreover, the method can be extended to graphs with weighted edges. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
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31 pages, 357 KiB  
Article
Numerical Linear Algebra for the Two-Dimensional Bertozzi–Esedoglu–Gillette–Cahn–Hilliard Equation in Image Inpainting
by Yahia Awad, Hussein Fakih and Yousuf Alkhezi
Mathematics 2023, 11(24), 4952; https://doi.org/10.3390/math11244952 - 14 Dec 2023
Cited by 1 | Viewed by 875
Abstract
In this paper, we present a numerical linear algebra analytical study of some schemes for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation. Both 1D and 2D finite difference discretizations in space are proposed with semi-implicit and implicit discretizations on time. We prove that our proposed numerical solutions [...] Read more.
In this paper, we present a numerical linear algebra analytical study of some schemes for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation. Both 1D and 2D finite difference discretizations in space are proposed with semi-implicit and implicit discretizations on time. We prove that our proposed numerical solutions converge to continuous solutions. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
14 pages, 322 KiB  
Article
On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations
by Ivan G. Ivanov and Hongli Yang
Mathematics 2023, 11(21), 4436; https://doi.org/10.3390/math11214436 - 26 Oct 2023
Cited by 1 | Viewed by 1343
Abstract
In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, [...] Read more.
In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X+A∗X−1A=Q. We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
12 pages, 311 KiB  
Article
Schur Complement-Based Infinity Norm Bound for the Inverse of Dashnic-Zusmanovich Type Matrices
by Wenlong Zeng, Jianzhou Liu and Hongmin Mo
Mathematics 2023, 11(10), 2254; https://doi.org/10.3390/math11102254 - 11 May 2023
Cited by 2 | Viewed by 1462
Abstract
It is necessary to explore more accurate estimates of the infinity norm of the inverse of a matrix in both theoretical analysis and practical applications. This paper focuses on obtaining a tighter upper bound on the infinite norm of the inverse of Dashnic–Zusmanovich-type [...] Read more.
It is necessary to explore more accurate estimates of the infinity norm of the inverse of a matrix in both theoretical analysis and practical applications. This paper focuses on obtaining a tighter upper bound on the infinite norm of the inverse of Dashnic–Zusmanovich-type (DZT) matrices. The realization of this goal benefits from constructing the scaling matrix of DZT matrices and the diagonal dominant degrees of Schur complements of DZT matrices. The effectiveness and superiority of the obtained bounds are demonstrated through several numerical examples involving random variables. Moreover, a lower bound for the smallest singular value is given by using the proposed bound. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
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28 pages, 559 KiB  
Article
Componentwise Perturbation Analysis of the QR Decomposition of a Matrix
by Petko H. Petkov
Mathematics 2022, 10(24), 4687; https://doi.org/10.3390/math10244687 - 10 Dec 2022
Cited by 4 | Viewed by 2401
Abstract
The paper presents a rigorous perturbation analysis of the QR decomposition A=QR of an n×m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and [...] Read more.
The paper presents a rigorous perturbation analysis of the QR decomposition A=QR of an n×m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and the subspaces spanned by the first p≤m columns of A. The new bounds are less conservative than the known bounds and are significantly better than the normwise bounds. An iterative scheme is proposed to determine global componentwise bounds in the case of perturbations for which such bounds are valid. Several numerical results are given that illustrate the analysis and the quality of the bounds obtained. Full article
(This article belongs to the Special Issue Numerical Analysis and Matrix Computations: Theory and Applications)
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