scholarly journals A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 893
Author(s):  
Yunlan Wei ◽  
Yanpeng Zheng ◽  
Zhaolin Jiang ◽  
Sugoog Shon
Keyword(s):  
Toeplitz Matrices ◽  
Schur Complement ◽  
Matrix Transformations ◽  
Matrix Decompositions ◽  
Mersenne Numbers

In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

Matrix Decompositions Involving the Schur Complement

1975 ◽  
Vol 28 (3) ◽  
pp. 577-587 ◽  
Author(s):  
David Carlson
Keyword(s):  
Schur Complement ◽  
Matrix Decompositions

Recursive self preconditioning method based on Schur complement for Toeplitz matrices

2012 ◽  
Vol 62 (3) ◽  
pp. 505-525 ◽  
Author(s):  
Faezeh Toutounian ◽  
Nasser Akhoundi
Keyword(s):  
Toeplitz Matrices ◽  
Schur Complement ◽  
Preconditioning Method

Analysis between graph-based and Power Transfer Distribution Factors (PTDF)-based model reduction methods in Electric Power Systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego A. Monroy-Ortiz ◽  
Sergio A. Dorado-Rojas ◽  
Eduardo Mojica-Nava ◽  
Sergio Rivera
Keyword(s):  
Power Systems ◽  
Power System ◽  
Model Reduction ◽  
Electrical Power ◽  
Schur Complement ◽  
Power Transfer ◽  
Electrical Power System ◽  
Admittance Matrix ◽  
Test Beds ◽  
Power Transfer Distribution Factors

Abstract This article presents a comparison between two different methods to perform model reduction of an Electrical Power System (EPS). The first is the well-known Kron Reduction Method (KRM) that is used to remove the interior nodes (also known as internal, passive, or load nodes) of an EPS. This method computes the Schur complement of the primitive admittance matrix of an EPS to obtain a reduced model that preserves the information of the system as seen from to the generation nodes. Since the primitive admittance matrix is equivalent to the Laplacian of a graph that represents the interconnections between the nodes of an EPS, this procedure is also significant from the perspective of graph theory. On the other hand, the second procedure based on Power Transfer Distribution Factors (PTDF) uses approximations of DC power flows to define regions to be reduced within the system. In this study, both techniques were applied to obtain reduced-order models of two test beds: a 14-node IEEE system and the Colombian power system (1116 buses), in order to test scalability. In analyzing the reduction of the test beds, the characteristics of each method were classified and compiled in order to know its advantages depending on the type of application. Finally, it was found that the PTDF technique is more robust in terms of the definition of power transfer in congestion zones, while the KRM method may be more accurate.

The inverses and eigenpairs of tridiagonal Toeplitz matrices with perturbed rows

2021 ◽  
Author(s):  
Yunlan Wei ◽  
Yanpeng Zheng ◽  
Zhaolin Jiang ◽  
Sugoog Shon
Keyword(s):  
Toeplitz Matrices

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

Polarimetry of soil and vegetation in the visible II: Mueller matrix decompositions

2021 ◽  
pp. 107622
Author(s):  
Sergey N. Savenkov ◽  
Alexander A. Kohkanovsky ◽  
Evgen A. Oberemok ◽  
Ivan S. Kolomiets ◽  
Alexander S. Klimov
Keyword(s):  
Mueller Matrix ◽  
Matrix Decompositions

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.

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