Matrix Decompositions Involving the Schur Complement

David Carlson

doi:10.1137/0128047

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

Mathematics ◽

10.3390/math7100893 ◽

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.

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

International Journal of Emerging Electric Power Systems ◽

10.1515/ijeeps-2019-0278 ◽

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.

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Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽

10.3390/e23020237 ◽

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.

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

BIT Numerical Mathematics ◽

10.1007/s10543-020-00838-9 ◽

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.

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

Journal of Quantitative Spectroscopy and Radiative Transfer ◽

10.1016/j.jqsrt.2021.107622 ◽

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

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Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

Journal of Numerical Mathematics ◽

10.1515/jnma-2020-0048 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Zdeněk Dostál ◽

Tomáš Brzobohatý ◽

Oldřich Vlach

Keyword(s):

Condition Number ◽

Three Dimensional ◽

Schur Complement ◽

Decomposition Methods ◽

Stiffness Matrices ◽

Spectral Bounds ◽

Schur Complements ◽

Linear Problems ◽

Primal Method ◽

Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

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