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Journal articles on the topic 'Diagonally dominant linear systems'

Author: Grafiati

Published: 6 September 2023

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1

Huang, Rong, Jianzhou Liu, and Li Zhu. "Accurate solutions of diagonally dominant tridiagonal linear systems." BIT Numerical Mathematics 54, no. 3 (March 18, 2014): 711–27. http://dx.doi.org/10.1007/s10543-014-0481-5.

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2

Zhang, Cheng-yi, Dan Ye, Cong-Lei Zhong, and SHUANGHUA SHUANGHUA. "Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices." Electronic Journal of Linear Algebra 30 (February 8, 2015): 843–70. http://dx.doi.org/10.13001/1081-3810.1972.

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It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible Hâmatrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general Hâmatrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general Hâmatrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general Hâmatrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

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3

Doan, T. S., and S. Siegmund. "Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/210156.

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We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity.

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4

Shahruz, S. M., and F. Ma. "Approximate Decoupling of the Equations of Motion of Linear Underdamped Systems." Journal of Applied Mechanics 55, no. 3 (September 1, 1988): 716–20. http://dx.doi.org/10.1115/1.3125855.

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One common procedure in the solution of a normalized damped linear system with small off-diagonal damping elements is to replace the normalized damping matrix by a selected diagonal matrix. The extent of approximation introduced by this method of decoupling the system is evaluated, and tight error bounds are derived. Moreover, if the normalized damping matrix is diagonally dominant, it is shown that decoupling the system by neglecting the off-diagonal elements indeed minimizes the error bound.

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5

Frommer, A., and G. Mayer. "Linear systems with Ω-diagonally dominant matrices and related ones." Linear Algebra and its Applications 186 (June 1993): 165–81. http://dx.doi.org/10.1016/0024-3795(93)90289-z.

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6

Siahlooei, Esmaeil, and Seyed Abolfazl Shahzadeh Fazeli. "Two Iterative Methods for Solving Linear Interval Systems." Applied Computational Intelligence and Soft Computing 2018 (October 8, 2018): 1–13. http://dx.doi.org/10.1155/2018/2797038.

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Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where Ã is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solves Ãx̃=b̃, where Ã is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution x̃ for linear interval systems, where Ãx̃=b̃; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.

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7

Spielman, Daniel A., and Shang-Hua Teng. "Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems." SIAM Journal on Matrix Analysis and Applications 35, no. 3 (January 2014): 835–85. http://dx.doi.org/10.1137/090771430.

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8

Wang, Guangbin, Hao Wen, and Ting Wang. "Convergence of GAOR Iterative Method with Strictly Diagonally Dominant Matrices." Journal of Applied Mathematics 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/713795.

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We discuss the convergence of GAOR method for linear systems with strictly diagonally dominant matrices. Moreover, we show that our results are better than ones of Darvishi and Hessari (2006), Tian et al. (2008) by using three numerical examples.

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9

Belhaj, Skander, Fahd Hcini, Maher Moakher, and Yulin Zhang. "A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems." Journal of Mathematical Chemistry 59, no. 3 (February 2, 2021): 757–74. http://dx.doi.org/10.1007/s10910-021-01217-7.

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10

Tian, Gui-Xian, Ting-Zhu Huang, and Shu-Yu Cui. "Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices." Journal of Computational and Applied Mathematics 213, no. 1 (March 2008): 240–47. http://dx.doi.org/10.1016/j.cam.2007.01.016.

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11

El-Kurdi, Yousef, Warren J. Gross, and Dennis Giannacopoulos. "Efficient Implementation of Gaussian Belief Propagation Solver for Large Sparse Diagonally Dominant Linear Systems." IEEE Transactions on Magnetics 48, no. 2 (February 2012): 471–74. http://dx.doi.org/10.1109/tmag.2011.2176318.

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12

Zaheer Ahmed, Zubair Ahmed Kalhoro, Abdul Wasim Shaikh, Muhammad Shakeel Rind Baloch, and Owais Ali Rajput. "An Improved Iterative Scheme using Successive Over-relaxation for Solution of Linear System of Equations." Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 59, no. 3 (August 21, 2022): 43–51. http://dx.doi.org/10.53560/ppasa(59-3)653.

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To solve the system of linear equations is one of the hottest topics in iterative methods. The system of linear equations occurs in business, engineering, social and in sensitive research areas like medicine, therefore applying efficient matrix solvers to such systems is crucial. In this paper, an improved iterative scheme using successive overrelaxation has been constructed. The proposed iterative method converges well when a linear system’s matrix is M-matrix, Symmetric positive definite with some conditions, irreducibly diagonally dominant, strictly diagonally dominant, and H-matrix. Such type of linear system of equations does arise usually from ordinary differential equations and partial differential equations. The improved iterative scheme has decreased spectral radius, improved stability and reduced the number of iterations. To show the effectiveness of the improved scheme, it is compared with the refinement of generalized successive over-relaxation and generalized successive over-relaxation method with the help of numerical experiments using MATLAB software.

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13

Darvishi, M. T., and P. Hessari. "On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices." Applied Mathematics and Computation 176, no. 1 (May 2006): 128–33. http://dx.doi.org/10.1016/j.amc.2005.09.051.

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14

Zhang, Cheng-yi, Zichen Xue, and Shuanghua Luo. "A convergence analysis of SOR iterative methods for linear systems with weak H-matrices." Open Mathematics 14, no. 1 (January 1, 2016): 747–60. http://dx.doi.org/10.1515/math-2016-0065.

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AbstractIt is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

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15

Enyew, Tesfaye Kebede, Gurju Awgichew, Eshetu Haile, and Gashaye Dessalew Abie. "Second-refinement of Gauss-Seidel iterative method for solving linear system of equations." Ethiopian Journal of Science and Technology 13, no. 1 (April 30, 2020): 1–15. http://dx.doi.org/10.4314/ejst.v13i1.1.

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Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

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16

Macintosh, Hamish J., Jasmine E. Banks, and Neil A. Kelson. "Implementing and Evaluating an Heterogeneous, Scalable, Tridiagonal Linear System Solver with OpenCL to Target FPGAs, GPUs, and CPUs." International Journal of Reconfigurable Computing 2019 (October 13, 2019): 1–13. http://dx.doi.org/10.1155/2019/3679839.

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Solving diagonally dominant tridiagonal linear systems is a common problem in scientific high-performance computing (HPC). Furthermore, it is becoming more commonplace for HPC platforms to utilise a heterogeneous combination of computing devices. Whilst it is desirable to design faster implementations of parallel linear system solvers, power consumption concerns are increasing in priority. This work presents the oclspkt routine. The oclspkt routine is a heterogeneous OpenCL implementation of the truncated SPIKE algorithm that can use FPGAs, GPUs, and CPUs to concurrently accelerate the solving of diagonally dominant tridiagonal linear systems. The routine is designed to solve tridiagonal systems of any size and can dynamically allocate optimised workloads to each accelerator in a heterogeneous environment depending on the accelerator’s compute performance. The truncated SPIKE FPGA solver is developed first for optimising OpenCL device kernel performance, global memory bandwidth, and interleaved host to device memory transactions. The FPGA OpenCL kernel code is then refactored and optimised to best exploit the underlying architecture of the CPU and GPU. An optimised TDMA OpenCL kernel is also developed to act as a serial baseline performance comparison for the parallel truncated SPIKE kernel since no FPGA tridiagonal solver capable of solving large tridiagonal systems was available at the time of development. The individual GPU, CPU, and FPGA solvers of the oclspkt routine are 110%, 150%, and 170% faster, respectively, than comparable device-optimised third-party solvers and applicable baselines. Assessing heterogeneous combinations of compute devices, the GPU + FPGA combination is found to have the best compute performance and the FPGA-only configuration is found to have the best overall estimated energy efficiency.

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17

Audu, KJ, YA Yahaya, KR Adeboye, and UY Abubakar. "Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems." NIGERIAN ANNALS OF PURE AND APPLIED SCIENCES 4, no. 1 (August 19, 2021): 53–61. http://dx.doi.org/10.46912/napas.226.

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Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

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18

Thompson, S. "Pseudo-Decoupling and Control of Unidentified Plant." Journal of Dynamic Systems, Measurement, and Control 108, no. 1 (March 1, 1986): 74–79. http://dx.doi.org/10.1115/1.3143746.

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A procedure based on measured step responses, is presented for the pseudo-decoupling of basically linear, nonoscillatory, multivariable plant. The method generates pre- and post-compensator matrices that will tend to make an otherwise nondominant plant diagonally dominant. Once this is achieved, each loop is then tuned on an individual basis using one of the classical on-line methods, thus eliminating the need for a model. The method is illustrated by means of two examples. The first demonstrates the design procedure and the second reports on the application of the method in controlling the power generated by an internal combustion engine.

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19

Duan, Ruirui, Junmin Li, Yanni Zhang, Ying Yang, and Guopei Chen. "Stability analysis and H∞ control of discrete T–S fuzzy hyperbolic systems." International Journal of Applied Mathematics and Computer Science 26, no. 1 (March 1, 2016): 133–45. http://dx.doi.org/10.1515/amcs-2016-0009.

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Abstract This paper focuses on the problem of constraint control for a class of discrete-time nonlinear systems. Firstly, a new discrete T–S fuzzy hyperbolic model is proposed to represent a class of discrete-time nonlinear systems. By means of the parallel distributed compensation (PDC) method, a novel asymptotic stabilizing control law with the “soft” constraint property is designed. The main advantage is that the proposed control method may achieve a small control amplitude. Secondly, for an uncertain discrete T–S fuzzy hyperbolic system with external disturbances, by the proposed control method, the robust stability and H∞ performance are developed by using a Lyapunov function, and some sufficient conditions are established through seeking feasible solutions of some linear matrix inequalities (LMIs) to obtain several positive diagonally dominant (PDD) matrices. Finally, the validity and feasibility of the proposed schemes are demonstrated by a numerical example and a Van de Vusse one, and some comparisons of the discrete T–S fuzzy hyperbolic model with the discrete T–S fuzzy linear one are also given to illustrate the advantage of our approach.

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20

De la Sen, M. "Stabilization of Continuous-Time Adaptive Control Systems with Possible Input Saturation through a Controllable Modified Estimation Model." Nonlinear Analysis: Modelling and Control 9, no. 1 (January 25, 2004): 3–37. http://dx.doi.org/10.15388/na.2004.9.1.15167.

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This paper presents an indirect adaptive control scheme for linear continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. An alternative modification scheme based on the achievement of a modified diagonally dominant Sylvester matrix of the parameter estimates is also proposed. This diagonal dominance is achieved through estimates modification as a way to guarantee the controllability of the modified estimated model when a controllability measure of the estimation model without modification fails. In both schemes, the use of a hysteresis switching function for the modification of the estimates is not required to ensure the controllability of the modified estimated model. Both schemes ensure that chattering due to switches associated with the modification is not present. The results are extended to the first-order case when the input is subject to saturation being modeled as a sigmoid function. In this case, a hysteresis-type switching law is used to implement the estimates modification.

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21

Liu, Jianzhou, Zhuohong Huang, Li Zhu, and Zejun Huang. "Theorems on Schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems." Linear Algebra and its Applications 435, no. 12 (December 2011): 3085–100. http://dx.doi.org/10.1016/j.laa.2011.05.023.

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22

Meng, Fanwei, Dini Wang, Penghui Yang, and Guanzhou Xie. "Application of Sum of Squares Method in Nonlinear H∞ Control for Satellite Attitude Maneuvers." Complexity 2019 (November 30, 2019): 1–10. http://dx.doi.org/10.1155/2019/5124108.

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The Hamilton–Jacobi–Issacs (HJI) inequality is the most basic relation in nonlinear H∞ design, to which no effective analytical solution is currently available. The sum of squares (SOS) method can numerically solve nonlinear problems that are not easy to solve analytically, but it still cannot solve HJI inequalities directly. In this paper, an HJI inequality suitable for SOS is firstly derived to solve the problem of nonconvex optimization. Then, the problems of SOS in nonlinear H∞ design are analyzed in detail. Finally, a two-step iterative design method for solving nonlinear H∞ control is presented. The first step is to design an adjustable nonlinear state feedback of the gain array of the system using SOS. The second step is to solve the L2 gain of the system; the optimization problem is solved by a graphical analytical method. In the iterative design, a diagonally dominant design idea is proposed to reduce the numerical error of SOS. The nonlinear H∞ control design of a polynomial system for large satellite attitude maneuvers is taken as our example. Simulation results show that the SOS method is comparable to the LMI method used for linear systems, and it is expected to find a broad range of applications in the analysis and design of nonlinear systems.

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23

Saul, Lawrence K. "A tractable latent variable model for nonlinear dimensionality reduction." Proceedings of the National Academy of Sciences 117, no. 27 (June 22, 2020): 15403–8. http://dx.doi.org/10.1073/pnas.1916012117.

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We propose a latent variable model to discover faithful low-dimensional representations of high-dimensional data. The model computes a low-dimensional embedding that aims to preserve neighborhood relationships encoded by a sparse graph. The model both leverages and extends current leading approaches to this problem. Like t-distributed Stochastic Neighborhood Embedding, the model can produce two- and three-dimensional embeddings for visualization, but it can also learn higher-dimensional embeddings for other uses. Like LargeVis and Uniform Manifold Approximation and Projection, the model produces embeddings by balancing two goals—pulling nearby examples closer together and pushing distant examples further apart. Unlike these approaches, however, the latent variables in our model provide additional structure that can be exploited for learning. We derive an Expectation–Maximization procedure with closed-form updates that monotonically improve the model’s likelihood: In this procedure, embeddings are iteratively adapted by solving sparse, diagonally dominant systems of linear equations that arise from a discrete graph Laplacian. For large problems, we also develop an approximate coarse-graining procedure that avoids the need for negative sampling of nonadjacent nodes in the graph. We demonstrate the model’s effectiveness on datasets of images and text.

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24

Jiang, Xu, Jing Yu Hua, and Qin Ling Zhang. "Research on Model Reference Robust Control for Multivariable Linear System." Advanced Materials Research 712-715 (June 2013): 2761–67. http://dx.doi.org/10.4028/www.scientific.net/amr.712-715.2761.

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This paper studies the output-feedback model reference robust control for MIMOlinear systems with generalized relative degree one. A new robust controlscheme is proposed within the framework of model reference control. Under theassumption that the high-frequency gain matrix of the plant can be transformedto a glass of main diagonal dominant matrix via full rank transformation, it isshown that all signals of the closed-loop system are globally uniformly boundedand meanwhile, the tracking errors converge to a residual set that can be madearbitrarily small by properly choosing some design parameters. Simulationresults are presented to illustrate the effectiveness of the proposed scheme.

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25

Jiang, Xu, and Zhi Tao Feng. "Research on Multivariable Model Reference Variable Structure Control." Applied Mechanics and Materials 281 (January 2013): 121–26. http://dx.doi.org/10.4028/www.scientific.net/amm.281.121.

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This paper studies the output-feedback variable structure control for MIMO (multi-input multi-output) linear systems with generalized relative degree one. A new variable structure control scheme is proposed within the framework of model reference control. Under the assumption that the high frequency gain matrix of the plant can be transformed to a main diagonal dominant matrix via full rank transformation, it is shown that all signals of the closed-loop system are globally uniformly bounded and meanwhile, the tracking error converges to zero exponentially. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

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26

Noei, Maziar, Paul Luckner, Tobias Linn, and Christoph Jungemann. "Numerical aspects of a Godunov-type stabilization scheme for the Boltzmann transport equation." Journal of Computational Electronics 21, no. 1 (January 18, 2022): 153–68. http://dx.doi.org/10.1007/s10825-021-01846-w.

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AbstractWe discuss the numerical aspects of the Boltzmann transport equation (BE) for electrons in semiconductor devices, which is stabilized by Godunov’s scheme. The k-space is discretized with a grid based on the total energy to suppress spurious diffusion in the stationary case. Band structures of arbitrary shape can be handled. In the stationary case, the discrete BE yields always nonnegative distribution functions and the corresponding system matrix has only eigenvalues with positive real parts (diagonally dominant matrix) resulting in an excellent numerical stability. In the transient case, this property yields an upper limit for the time step ensuring the stability of the CPU-efficient forward Euler scheme and a positive distribution function. Similar to the Monte-Carlo (MC) method, the discrete BE can be solved in time together with the Poisson equation (PE), where the time steps for the PE are split into shorter time steps for the BE, which can be performed at minor additional computational cost. Thus, similar to the MC method, the transient approach is matrix-free and the solution of memory and CPU intensive large systems of linear equations is avoided. The numerical properties of the approach are demonstrated for a silicon nanowire NMOSFET, for which the stationary I–V characteristics, small-signal admittance parameters and the switching behavior are simulated with and without strong scattering. The spurious damping introduced by Godunov’s (upwind) scheme is found to be negligible in the technically relevant frequency range. The inherent asymmetry of the upwind scheme results in an error for very strong scattering that can be alleviated by a finer grid in transport direction.

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27

ARGOUN, M. B. "Model reduction of diagonally dominant systems." International Journal of Control 43, no. 3 (March 1986): 819–35. http://dx.doi.org/10.1080/00207178608933505.

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28

Forsgren, Anders. "On Linear Least-Squares Problems with Diagonally Dominant Weight Matrices." SIAM Journal on Matrix Analysis and Applications 17, no. 4 (October 1996): 763–88. http://dx.doi.org/10.1137/s0895479895284014.

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29

Ammar, Aymen, Slim Fakhfakh, and Aref Jeribi. "Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations." Journal of Pseudo-Differential Operators and Applications 7, no. 4 (March 25, 2016): 493–509. http://dx.doi.org/10.1007/s11868-016-0154-z.

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30

Sun, Deshu, and Feng Wang. "New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices." Open Mathematics 15, no. 1 (July 21, 2017): 978–86. http://dx.doi.org/10.1515/math-2017-0080.

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Abstract Some new error bounds for the linear complementarity problems are obtained when the involved matrices are weakly chained diagonally dominant B-matrices. Numerical examples are given to show the effectiveness of the proposed bounds.

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31

Baloch, Muhammad Shakeel Rind, Zubair Ahmed Kalhoro, Mir Sarfraz Khalil, and Prof Abdul Wasim Shaikh. "A New Improved Classical Iterative Algorithm for Solving System of Linear Equations." Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 58, no. 4 (March 28, 2022): 69–81. http://dx.doi.org/10.53560/ppasa(58-4)638.

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The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

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32

McNally, Jeffrey Mark. "A fast algorithm for solving diagonally dominant symmetric pentadiagonal Toeplitz systems." Journal of Computational and Applied Mathematics 234, no. 4 (June 2010): 995–1005. http://dx.doi.org/10.1016/j.cam.2009.03.001.

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33

Morzfeld, M., N. Ajavakom, and F. Ma. "Diagonal dominance of damping and the decoupling approximation in linear vibratory systems." Journal of Sound and Vibration 320, no. 1-2 (February 2009): 406–20. http://dx.doi.org/10.1016/j.jsv.2008.07.025.

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34

Matcovschi, Mihaela-Hanako, and Octavian Pastravanu. "Diagonally invariant exponential stability and stabilizability of switching linear systems." Mathematics and Computers in Simulation 82, no. 8 (April 2012): 1407–18. http://dx.doi.org/10.1016/j.matcom.2011.07.011.

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35

Sun, Deshu. "Note on error bounds for linear complementarity problems involving $ B^S $-matrices." AIMS Mathematics 7, no. 2 (2022): 1896–906. http://dx.doi.org/10.3934/math.2022109.

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<abstract><p>Using the range for the infinity norm of inverse matrix of a strictly diagonally dominant $ M $-matrix, some new error bounds for the linear complementarity problem are obtained when the involved matrix is a $ B^S $-matrix. Theory analysis and numerical examples show that these upper bounds are more accurate than some existing results.</p></abstract>

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36

Riedel, Kurt S. "Block diagonally dominant positive definite approximate filters and smoothers." Automatica 29, no. 3 (May 1993): 779–83. http://dx.doi.org/10.1016/0005-1098(93)90074-4.

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37

Hu, Jin-Xiu, Xiao-Wei Gao, Zhi-Chao Yuan, Jian Liu, and Shi-Zhang Huang. "SEBSM-Based Iterative Method for Solving Large Systems of Linear Equations and Its Applications in Engineering Computation." International Journal of Computational Methods 13, no. 05 (August 31, 2016): 1650024. http://dx.doi.org/10.1142/s0219876216500249.

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In this paper, a new iterative method, for solving sparse nonsymmetrical systems of linear equations is proposed based on the Simultaneous Elimination and Back-Substitution Method (SEBSM), and the method is applied to solve systems resulted in engineering problems solved using Finite Element Method (FEM). First, SEBSM is introduced for solving general linear systems using the direct method. And, then an iterative method based on SEBSM is presented. In the method, the coefficient matrix [Formula: see text] is split into lower, diagonally banded and upper matrices. The iterative convergence can be controlled by selecting a suitable bandwidth of the diagonally banded matrix. And the size of the working array needing to be stored in iteration is as small as the bandwidth of the diagonally banded matrix. Finally, an accelerating strategy for this iterative method is proposed by introducing a relaxation factor, which can speed up the convergence effectively if an optimal relaxation factor is chosen. Two numerical examples are given to demonstrate the behavior of the proposed method.

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Li, Chaoqian, and Yaotang Li. "Weakly chained diagonally dominant B-matrices and error bounds for linear complementarity problems." Numerical Algorithms 73, no. 4 (March 21, 2016): 985–98. http://dx.doi.org/10.1007/s11075-016-0125-8.

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Ito, Kazufumi, and Karl Kunisch. "Convergence of the Primal‐Dual Active Set Strategy for Diagonally Dominant Systems." SIAM Journal on Control and Optimization 46, no. 1 (January 2007): 14–34. http://dx.doi.org/10.1137/050632713.

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Matcovschi, Mihaela-Hanako, and Octavian Pastravanu. "Further Results on Diagonally Invariant Exponential Stability of Switching Linear Systems." Mathematical Problems in Engineering 2018 (July 25, 2018): 1–13. http://dx.doi.org/10.1155/2018/9419514.

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The concept of diagonally invariant exponential stability (DIES) was originally introduced for single-model linear systems and subsequently expanded in the study of linear systems with interval-type uncertainties and linear systems with arbitrary switching. The results presented in this article refer to new approaches to DIES characterization for arbitrary switching systems, which exploit mathematical tools completely different from earlier work. The previous papers are based on the properties of matrix norms and measures applied to the constituent matrices defining the switching system, while the present paper uses the eigenvalues and eigenvectors of the column and row representatives built for a set of matrices derived from the constituent matrices of the switching system. The applicability of previous and new results, respectively, is illustrated by case studies (in both continuous- and discrete-time) that lead to relevant comparisons between the two classes of analysis methods.

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Ismail, S., A. A. Pashilkar, R. Ayyagari, and N. Sundararajan. "Diagonally dominant backstepping autopilot for aircraft with unknown actuator failures and severe winds." Aeronautical Journal 118, no. 1207 (September 2014): 1009–38. http://dx.doi.org/10.1017/s0001924000009726.

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Abstract A novel formulation of the flight dynamic equations is presented that permits a rapid solution for the design of trajectory following autopilots for nonlinear aircraft dynamic models. A robust autopilot control structure is developed based on the combination of the good features of the nonlinear dynamic inversion (NDI) method, integrator backstepping method, time scale separation and control allocation methods. The aircraft equations of motion are formulated in suitable variables so that the matrices involved in the block backstepping control design method are diagonally dominant. This allows us to use a linear controller structure for a trajectory following autopilot for the nonlinear aircraft model using the well known loop by loop controller design approach. The resulting autopilot for the fixed-wing rigid-body aircraft with a cascaded structure is referred to as the diagonally dominant backstepping (DDBS) controller. The method is illustrated here for an aircraft auto-landing problem under unknown actuator failures and severe winds. The requirement of state and control surface limiting is also addressed in the context of the design of the DDBS controller.

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Berger, Guillaume O., and Raphaël M. Jungers. "p-dominant switched linear systems." Automatica 132 (October 2021): 109801. http://dx.doi.org/10.1016/j.automatica.2021.109801.

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Liu, Yebo Xiong &. Jianzhou. "Norm Estimates for the Inverses of Strictly Diagonally Dominant $M$-Matrices and Linear Complementarity Problems." East Asian Journal on Applied Mathematics 11, no. 3 (June 2021): 487–514. http://dx.doi.org/10.4208/eajam.210820.161120.

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井, 霞. "A New Error Bound for Linear Complementarity Problems for Weakly Chained Diagonally Dominant B-Matrices." Advances in Applied Mathematics 06, no. 07 (2017): 850–56. http://dx.doi.org/10.12677/aam.2017.67102.

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Zhao, Ruijuan, Bing Zheng, and Maolin Liang. "A new error bound for linear complementarity problems with weakly chained diagonally dominant B-matrices." Applied Mathematics and Computation 367 (February 2020): 124788. http://dx.doi.org/10.1016/j.amc.2019.124788.

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Dessalew, Gashaye, Tesfaye Kebede, Gurju Awgichew, and Assaye Walelign. "Refinement of Multiparameters Overrelaxation (RMPOR) Method." Journal of Mathematics 2021 (August 23, 2021): 1–10. http://dx.doi.org/10.1155/2021/2804698.

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In this paper, we present refinement of multiparameters overrelaxation (RMPOR) method which is used to solve the linear system of equations. We investigate its convergence properties for different matrices such as strictly diagonally dominant matrix, symmetric positive definite matrix, and M-matrix. The proposed method minimizes the number of iterations as compared with the multiparameter overrelaxation method. Its spectral radius is also minimum. To show the efficiency of the proposed method, we prove some theorems and take some numerical examples.

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Chen, Minghua, Sven-Erik Ekström, and Stefano Serra-Capizzano. "A Multigrid Method for Nonlocal Problems: Non--Diagonally Dominant or Toeplitz-Plus-Tridiagonal Systems." SIAM Journal on Matrix Analysis and Applications 41, no. 4 (January 2020): 1546–70. http://dx.doi.org/10.1137/18m1210460.

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Bellavia, Stefania, Valentina De Simone, Daniela di Serafino, and Benedetta Morini. "A Preconditioning Framework for Sequences of Diagonally Modified Linear Systems Arising in Optimization." SIAM Journal on Numerical Analysis 50, no. 6 (January 2012): 3280–302. http://dx.doi.org/10.1137/110860707.

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Vepakomma, Praneeth, and Ahmed Elgammal. "A fast algorithm for manifold learning by posing it as a symmetric diagonally dominant linear system." Applied and Computational Harmonic Analysis 40, no. 3 (May 2016): 622–28. http://dx.doi.org/10.1016/j.acha.2015.10.004.

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Glüge, R., H. Altenbach, and S. Eisenträger. "Locally-synchronous, iterative solver for Fourier-based homogenization." Computational Mechanics 68, no. 3 (February 16, 2021): 599–618. http://dx.doi.org/10.1007/s00466-021-01975-w.

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AbstractWe use the algebraic orthogonality of rotation-free and divergence-free fields in the Fourier space to derive the solution of a class of linear homogenization problems as the solution of a large linear system. The effective constitutive tensor constitutes only a small part of the solution vector. Therefore, we propose to use a synchronous and local iterative method that is capable to efficiently compute only a single component of the solution vector. If the convergence of the iterative solver is ensured, i.e., the system matrix is positive definite and diagonally dominant, it outperforms standard direct and iterative solvers that compute the complete solution. It has been found that for larger phase contrasts in the homogenization problem, the convergence is lost, and one needs to resort to other linear system solvers. Therefore, we discuss the linear system’s properties and the advantages as well as drawbacks of the presented homogenization approach.

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