Zbigniew I. Woźnicki · International Journal of Mathematics and Mathematical Sciences · 2001
The systematic analysis of convergence conditions, used in comparison theorems proven for different matrix splittings, is presented. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrix A = M 1 − N 1 = M 2 − N 2 . An equivalence of some conditions as well as an autonomous character of the conditions and A −1 N 2 ≥ A −1 N 1 ≥ 0 are pointed out. The secondary goal is to discuss some essential topics related with existing comparison theorems.
Cited by 25Open access
Shengxin Zhu, Tong-Xiang Gu, Xingping Liu · Mathematical Foundations of Computing · 2020
For linear mixed models with co-variance matrices which are not linearly dependent on variance component parameters, we prove that the average of the observed information and the Fisher information can be split into two parts. The essential part enjoys a simple and computational friendly formula, while the other part which involves a lot of computations is a random zero matrix and thus is negligible.
Cited by 2Open access
RIZZARDI, CLARA, MELATO, MAURO · ArTS Archivio della ricerca di Trieste (University of Trieste https://www.units.it/) · 2003
Cited by 0
István Faragó, P. Tarvainen · Computers & Mathematics with Applications · 2001
Cited by 3
Ira W. Levin, Stanley Abramowitz · Chemical Physics Letters · 1971
Cited by 13
Zbigniew I. Woźnicki · 2001
Abstract. The systematic analysis of convergence conditions, used in comparison theo-rems proven for different matrix splittings, is presented. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrix A =M1−N1 =M2−N2. An equivalence of some conditions as well as an autonomous character of the conditions M−11 ≥M−12 ≥ 0 and A−1N2 ≥ A−1N1 ≥ 0 are pointed out. The secondary goal is to discuss some essential topics related with existing comparison theorems.
Cited by 0
Shoudong Shi, Demin Xu, Youming Li · 2009
There have been a number of variant Simultaneous Localization and Mapping (SLAM) algorithms which have made substantial progress towards large-area scalability by parameterizing the SLAM posterior within the information (canonical /inverse covariance) form. The most well-known and popular approach is the Sparse Extended Information Filter (SEIF) by Thrun et al. SEIF has been successfully implemented with a variety of challenging real-world data sets and has lead to new insights into scalable SLAM. The paper presents a new approach based upon tri-diagonal matrix splitting for solving the computational complexity problem of inverse information matrix. In this paper, by analyzing the every steps of information matrix update in SEIF process, we find that the most computational cost is in recovery information matrix (inverse matrix). According to this characteristics which the normalized information matrix exhibits a natural sparseness and the many of the off-diagonal elements are relatively weak (nearly to zero) in information matrix, we enforce the elements into zero according to setting a threshold and computer inverse information matrix with the method of tri-diagonal matrix splitting. As the new processing does not need direct matrix inversion, the computational complexity is much lower. The Computer simulation results indicate that SEIF process is faster and localization precision is no influence.
Cited by 0
Gábor Ivanyos, Lajos Rónyai, Josef Schicho · Journal of Algebra · 2012
Cited by 1Open access
Ganzhao Yuan, Wei‐Shi Zheng, Bernard Ghanem · 2017
Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization and cardinality regularized optimization as special cases. This paper proposes and analyzes a new Matrix Splitting Method (MSM) for minimizing composite functions. It can be viewed as a generalization of the classical Gauss-Seidel method and the Successive Over-Relaxation method for solving linear systems in the literature. Incorporating a new Gaussian elimination procedure, the matrix splitting method achieves state-of-the-art performance. For convex problems, we establish the global convergence, convergence rate, and iteration complexity of MSM, while for non-convex problems, we prove its global convergence. Finally, we validate the performance of our matrix splitting method on two particular applications: nonnegative matrix factorization and cardinality regularized sparse coding. Extensive experiments show that our method outperforms existing composite function minimization techniques in term of both efficiency and efficacy.
Cited by 1
Anna Neufeld · 2023
Implements the count splitting methodology from Neufeld et al. (2022) <<a href="https://doi.org/10.1093%2Fbiostatistics%2Fkxac047" target="_top">doi:10.1093/biostatistics/kxac047</a>> and Neufeld et al. (2023) <<a href="https://doi.org/10.48550/arXiv.2307.12985" target="_top">doi:10.48550/arXiv.2307.12985</a>>. Intended for turning a matrix of single-cell RNA sequencing counts, or similar count datasets, into independent folds that can be used for training/testing or cross validation. Assumes that the entries in the matrix are from a Poisson or a negative binomial distribution.
Cited by 0Open access
M. Anwar Hasan, Christophe Négre · IEEE Transactions on Computers · 2012
Computing the product of a Toeplitz matrix and a vector arises in various applications including cryptography. In this paper, we consider Toeplitz matrices and vectors with entries in $({\hbox{\rlap{I}\kern 2.0pt{\hbox{F}}}}_2)$. For improved efficiency in such computations, large Toeplitz matrices and vectors are recursively split and special formulas with subquadratic arithmetic complexity are applied. To this end, we first present a formula for the five-way splitting and then provide a generalization for the $(k)$-way splitting, where $(k)$ is an arbitrary integer. These formulas can be used to compute a Toeplitz matrix-vector product (TMVP) of size $(n)$ with an arithmetic complexity of $(O(n^{\log_k(k(k+1)/2)}))$.
Cited by 31
Koichi OZAKI, Takeshi Ogita, Shin’ichi Oishi · 2011
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Rónyai, Lajos, Ivanyos, Gábor, Schicho, J · SZTAKI Publication Repository (Hungarian Academy of Sciences) · 2011
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Andy Wathen · Linear Algebra and its Applications · 2025
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Pierre‐Luc Bacon, Doina Precup · arXiv (Cornell University) · 2016
We show that the Bellman operator underlying the options framework leads to a matrix splitting, an approach traditionally used to speed up convergence of iterative solvers for large linear systems of equations. Based on standard comparison theorems for matrix splittings, we then show how the asymptotic rate of convergence varies as a function of the inherent timescales of the options. This new perspective highlights a trade-off between asymptotic performance and the cost of computation associated with building a good set of options.
Cited by 1Open access
Susan Ghaderi, Hulda S. Haraldsdóttir, Masoud Ahookhosh et al. · Journal of Theoretical Biology · 2020
Characterising biochemical reaction network structure in mathematical terms enables the inference of functional biochemical consequences from network structure with existing mathematical techniques and spurs the development of new mathematics that exploits the peculiarities of biochemical network structure. The structure of a biochemical network may be specified by reaction stoichiometry, that is, the relative quantities of each molecule produced and consumed in each reaction of the network. A biochemical network may also be specified at a higher level of resolution in terms of the internal structure of each molecule and how molecular structures are transformed by each reaction in a network. The stoichiometry for a set of reactions can be compiled into a stoichiometric matrix N∈Z<sup>m×n</sup>, where each row corresponds to a molecule and each column corresponds to a reaction. We demonstrate that a stoichiometric matrix may be split into the sum of m-rank(N) moiety transition matrices, each of which corresponds to a subnetwork accessible to a structurally identifiable conserved moiety. The existence of this moiety matrix splitting is a property that distinguishes a stoichiometric matrix from an arbitrary rectangular matrix.
Cited by 7Open access
Vadym Slyusar · Cybernetics and Systems Analysis · 1999
Cited by 2
Manira Elena Narvaez-Saucedo, C.A. Lopez-Villalobos, Manuel A. Ramírez-Cabrera et al. · Computers & Fluids · 2025
• Introduces a meshless Monte Carlo method for natural convection. • Validated against finite-volume method with strong accuracy and scalability. • Transport in complex domains captured simply by discarding walkers. This work introduces a novel Markov Matrix Iterative Splitting Algorithm (MISA), a probabilistic operator splitting Monte Carlo approach designed for simulating natural convection in incompressible flows within the continuum regime. This approach effectively captures the nonlinear convective transport and stochastic diffusion while enforcing global incompressibility, without requiring mesh generation or matrix inversions common to deterministic solvers. Variance reduction techniques and a Multi-Level Monte Carlo framework improve convergence and stability. MISA naturally handles complex geometries with internal obstacles, incorporated effortlessly by excluding walkers inside them and imposing boundary conditions on their surfaces. Benchmark natural convection simulations in square cavities demonstrate MISA’s accuracy and computational efficiency compared to traditional finite-volume solvers. The method reliably resolves convective flow structures and temperature fields, highlighting its promising potential for complex fluid-thermal transport problems in irregular domains
Cited by 0Open access
D. S. Jones · Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences · 1991
Abstract Explicit expressions for the Wiener-Hopf factors of a 2 x 2 matrix are determined in terms of a single scalar. Examples of the application to various matrices are given.
Cited by 16
Zhengyi Zhang, Ahmed Sameh · Journal of Computational and Applied Mathematics · 2017
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