Article
Statistics & Probability
M. Gregoratti, D. Maran
Summary: This article introduces a new method for studying general probability distributions on R-n, and generalizes some results about the least singular value and the condition number of random matrices with i.i.d. Gaussian entries to the whole class of random matrices with i.i.d. rows.
STATISTICS & PROBABILITY LETTERS
(2021)
Article
Mathematics, Applied
Massimiliano Fasi, Nicholas J. Higham
Summary: The commonly used test matrix form is the randsvd matrix, which is constructed as the product A = U Sigma V*, where U and V are random orthogonal or unitary matrices from the Haar distribution and Sigma is a diagonal matrix of singular values. However, this construction is expensive at extreme scale and unsuitable for distributed memory environments due to the significant communication required. By relaxing the requirements on U and V, new algorithms have been derived that are more cost-effective and communication-efficient, particularly for generating matrices with specified 2-norm condition numbers. Numerical experiments have demonstrated the excellent efficiency and scalability of these algorithms.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Achiya Dax
Summary: In this paper, the behavior of the smallest singular values and condition numbers of a matrix with varying number of rows is examined. It is found that the smallest singular value sequence exhibits an anomaly, while the condition number sequence has a different characteristic. These findings are important for iterative methods in solving large linear systems.
Article
Computer Science, Theory & Methods
Vishesh Jain, Ashwin Sah, Mehtaab Sawhney
Summary: This study investigates random symmetric matrices of nxn dimensions and provides bounds on the probability of singularity. The results improve previous findings and introduce new concepts of arithmetic structure, suggesting a potential geometric approach to the study of the singular spectrum of symmetric random matrices.
COMBINATORICS PROBABILITY & COMPUTING
(2022)
Article
Mathematics, Applied
Minghua Lin, Mengyan Xie
Summary: In this note, we propose a simple idea to improve the lower bound for the smallest singular value of matrices, which is shown to be close to the exact value through numerical examples. We compare our bound with existing ones and discuss potential improvements in this area.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Xiao Shan Chen, Seak-Weng Vong
Summary: This paper discusses the sensitivity measures of multiple nonzero finite generalized singular values of two matrices with the same number of columns. The results show that multiple nonzero generalized singular values generally have multiple condition numbers, with their explicit expressions derived. A numerical example is provided to validate the results.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Engineering, Multidisciplinary
Zhengliu Zhou, Scott Keller
Summary: The LSFEM method has desirable properties, but may show unsatisfactory accuracy with low-order elements. A technique is proposed in this report to optimize the condition number of the stiffness matrix, significantly improving convergence properties. This technique can be applied to different problems and improve computational efficiency of the elemental stiffness matrix.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Interdisciplinary Applications
Gerardo Barrera, Paulo Manrique-Miron
Summary: This manuscript studies the distribution of singular values for random circulant matrices with Lyapunov condition. It shows that as the matrix dimension tends to infinity, the joint distribution of the extremal singular values converges to the product of Rayleigh and Gumbel laws.
Article
Psychology, Multidisciplinary
You-Lin Chen, Li-Jen Weng
Summary: Parallel analysis is a useful method proposed by Horn to determine the number of factors. Our study demonstrates that the distribution of eigenvalues closely resembles the Marcenko-Pastur distribution under finite datasets, while the distribution of L-1 deviates from the Tracy-Widom distribution. These findings support Horn's idea of using average eigenvalues from generated random data to reflect sampling fluctuations in finite datasets.
CURRENT PSYCHOLOGY
(2023)
Article
Computer Science, Interdisciplinary Applications
Halisson Alberdan Cavalcanti Cardoso, Silvio de Barros Melo, Ricardo Martins de Abreu Silva, Sidartha Azevedo Lobo de Carvalho, Silas Garrido Teixeira de Carvalho Santos, Carlos Costa Dantas
Summary: This study focuses on using the metaheuristic Greedy Randomized Adaptation Search Procedure (GRASP) to estimate the percentage counts of constituents of a compound represented by prompt gamma ray spectra, based on minimizing the condition number of the covariance matrices derived from the underlying linear system. By modifying the steps of GRASP, the research achieved successful results in estimating the compound fractions with prompt gamma ray spectra.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Computer Science, Information Systems
Pasan Dissanayake, Prathapasinghe Dharmawansa, Yang Chen
Summary: This paper investigates the distribution of random quantities related to eigenvalues of random matrices with specific covariance structures. Exact expressions for the probability density functions are derived using orthogonal polynomial approach, revealing the scaling behavior of the random quantities as matrix dimensions grow large.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Statistics & Probability
Ziliang Che, Patrick Lopatto
Summary: The paper investigates the limiting distribution properties of the least singular value of a random matrix with a specific structure. Under weak conditions on X and Y, it is proved that the distribution of the least singular value of this matrix, suitably rescaled, is the same as the distribution of the least singular value of a matrix of i.i.d. Gaussian random variables. The proof is based on a dynamical method previously used to study the local spectral statistics of sums of Hermitian matrices by Che and Landon.
ELECTRONIC JOURNAL OF PROBABILITY
(2021)
Article
Mathematics, Applied
Wushuang Liu, Xingkai Hu, Yaoqun Wang, Yuan Yi
Summary: In this paper, two lower bounds for the smallest singular value of non-singular matrices are presented. Additionally, numerical examples are used to demonstrate that these bounds are superior to existing ones.
JOURNAL OF MATHEMATICAL INEQUALITIES
(2023)
Article
Mathematics
Chaojun Yang
Summary: This paper examines singular value inequalities for sector matrices involving operator concave function and presents unitarily invariant norm inequalities for sector matrices.
JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Wasim Audeh
Summary: This paper discusses some singular value inequalities for compact operators and matrices on a complex separable Hilbert space.
ANNALS OF FUNCTIONAL ANALYSIS
(2022)