Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting in LU Factorization

Background: The 5G will lead to a great transformation in the mobile telecommunications sector. Objective: The huge challenges being faced by wireless communications such as the increased number of users have given a chance for 5G systems to be developed and considered as an alternative solution. The 5G technology will provide a higher data rate, reduced latency, more efficient power than the previous generations, higher system capacity, and more connected devices. Method: It will offer new different technologies and enhanced versions of the existing ones, as well as new features. 5G systems are going to use massive MIMO (mMIMO), which is a promising technology in the development of these systems. Furthermore, mMIMO will increase the wireless spectrum efficiency and improve the network coverage. Result: In this paper we present a brief survey on 5G and its technologies, discuss the mMIMO technology with its features and advantages, review the mMIMO capacity and energy efficiency and also presents the recent beamforming techniques. Conclusion: Finally, simulation of adopting different mMIMO detection algorithms are presented, which shows the alternating direction method of multipliers (ADMM)-based infinity-norm (ADMIN) detector has the best performance.

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Least singular value and condition number of a square random matrix with i.i.d. rows

Statistics & Probability Letters ◽

10.1016/j.spl.2021.109070 ◽

2021 ◽

pp. 109070

Author(s):

M. Gregoratti ◽

D. Maran

Keyword(s):

Condition Number ◽

Random Matrix ◽

Singular Value ◽

Least Singular Value

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Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

Special Matrices ◽

10.1515/spma-2020-0004 ◽

2020 ◽

Vol 8 (1) ◽

pp. 22-35

Author(s):

M. Shakil ◽

M. Ahsanullah

Keyword(s):

Order Statistics ◽

Condition Number ◽

Record Values ◽

Distributional Properties ◽

Upper Record Values ◽

Upper Record

AbstractThe objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

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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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