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Relevant bibliographies by topics / Gauss-Seidel iterative method

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Academic literature on the topic 'Gauss-Seidel iterative method'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Gauss-Seidel iterative method"

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Duan, Ban Xiang, Wen Ying Zeng, and Xiao Ping Zhu. "A Preconditioned Gauss-Seidel Iterative Method for Linear Complementarity Problem in Intelligent Materials System." Advanced Materials Research 340 (September 2011): 3–8. http://dx.doi.org/10.4028/www.scientific.net/amr.340.3.

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In this paper, the authors first set up new preconditioned Gauss-Seidel iterative method for solving the linear complementarity problem, whose preconditioned matrix is introduced. Then certain elementary operations row are performed on system matrix before applying the Gauss-Seidel iterative method. Moreover the sufficient conditions for guaranteeing the convergence of the new preconditioned Gauss-Seidel iterative method are presented. Lastly we report some computational results with the proposed method.

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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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黄, 江玲. "Convergence of Preconditioned Gauss-Seidel Iterative Method." Pure Mathematics 10, no. 11 (2020): 1007–13. http://dx.doi.org/10.12677/pm.2020.1011119.

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Akram, Muhammad, Ghulam Muhammad, Ali N. A. Koam, and Nawab Hussain. "Iterative Methods for Solving a System of Linear Equations in a Bipolar Fuzzy Environment." Mathematics 7, no. 8 (August 9, 2019): 728. http://dx.doi.org/10.3390/math7080728.

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We develop the solution procedures to solve the bipolar fuzzy linear system of equations (BFLSEs) with some iterative methods namely Richardson method, extrapolated Richardson (ER) method, Jacobi method, Jacobi over-relaxation (JOR) method, Gauss–Seidel (GS) method, extrapolated Gauss-Seidel (EGS) method and successive over-relaxation (SOR) method. Moreover, we discuss the properties of convergence of these iterative methods. By showing the validity of these methods, an example having exact solution is described. The numerical computation shows that the SOR method with ω = 1 . 25 is more accurate as compared to the other iterative methods.

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Zou, Youyi Jiang Limin. "Convergence of The Gauss-Seidel Iterative Method." Procedia Engineering 15 (2011): 1647–50. http://dx.doi.org/10.1016/j.proeng.2011.08.307.

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Liu, Zhongyun, Xiaorong Qin, Nianci Wu, and Yulin Zhang. "The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices." Canadian Mathematical Bulletin 60, no. 4 (December 1, 2017): 807–15. http://dx.doi.org/10.4153/cmb-2016-077-5.

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AbstractIt is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., T = C−S with C a circulantmatrix and S a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.

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Greenberg, Albert G., and Robert J. Vanderbei. "Quicker Convergence for Iterative Numerical Solutions to Stochastic Problems: Probabilistic Interpretations, Ordering Heuristics, and Parallel Processing." Probability in the Engineering and Informational Sciences 4, no. 4 (October 1990): 493–521. http://dx.doi.org/10.1017/s0269964800001790.

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Gauss-Seidel is a general method for solving a system of equations (possibly nonlinear). It makes repeated sweeps through the variables; within a sweep as each new estimate for a variable is computed, the current estimate for that variable is replaced with the new estimate immediately, instead of on completion of the sweep. The idea is to use new data as soon as it is computed. Gauss- Seidel is often efficient for computing the invariant measure of a Markov chain (especially if the transition matrix is sparse), and for computing the value function in optimal control problems. In many applications the computation can be significantly improved by appropriately ordering the variables within each sweep. A simple heuristic is presented here for computing an ordering that quickens convergence. In parallel processing, several variables must be computed simultaneously, which appears to work against Gauss-Seidel. Simple asynchronous parallel Gauss-Seidel methods are presented here. Experiments indicate that the methods retain the benefit of a good ordering, while further speeding up convergence by a factor of P if P processors participate.In this paper, we focus on the optimal stopping problem. A probabilistic interpretation of the Gauss-Seidel (and the Jacobi) method for computing the value function is given, which motivates our ordering heuristic. However, the ordering heuristic and parallel processing methods apply in a broader context, in particular, to the important problem of computing the invariant measure of a Markov chain.

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Pu, Bingyuan, and Xun Yuan. "The alternate iterative Gauss-Seidel method for linear systems." Journal of Physics: Conference Series 1411 (November 2019): 012008. http://dx.doi.org/10.1088/1742-6596/1411/1/012008.

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Zhang, Shi Guang, and Ting Zhou. "Convergence Results of the Improving Modified Gauss-Seidel (IMGS) Iterative Method for a Symmetric Positive Definite Matrix." Advanced Materials Research 989-994 (July 2014): 1794–97. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.1794.

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In this paper, in order to improve the convergence rates of iterative method solving the linear system, the improving modified Gauss-Seidel (IMGS) iterative method with a preconditioner is proposed. Some convergence and comparison results are given when is a symmetric definite matrix are provided.

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Wu, Shiliang, and Tingzhu Huang. "A modified AOR-type iterative method for L-matrix linear systems." ANZIAM Journal 49, no. 2 (October 2007): 281–92. http://dx.doi.org/10.1017/s1446181100012840.

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AbstractBoth Evans et al. and Li et al. have presented preconditioned methods for linear systems to improve the convergence rates of AOR-type iterative schemes. In this paper, we present a new preconditioner. Some comparison theorems on preconditioned iterative methods for solving L-matrix linear systems are presented. Comparison results and a numerical example show that convergence of the preconditioned Gauss-Seidel method is faster than that of the preconditioned AOR iterative method.

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Dissertations / Theses on the topic "Gauss-Seidel iterative method"

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Liang, Yunxu. "Improved Gauss-Seidel iterative method on power networks." [Gainesville, Fla.] : University of Florida, 2004. http://purl.fcla.edu/fcla/etd/UFE0008140.

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Wu, Wei. "Paving the Randomized Gauss-Seidel." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/scripps_theses/1074.

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The Randomized Gauss-Seidel Method (RGS) is an iterative algorithm that solves overdetermined systems of linear equations Ax = b. This paper studies an update on the RGS method, the Randomized Block Gauss-Seidel Method. At each step, the algorithm greedily minimizes the objective function L(x) = kAx bk2 with respect to a subset of coordinates. This paper describes a Randomized Block Gauss-Seidel Method (RBGS) which uses a randomized control method to choose a subset at each step. This algorithm is the first block RGS method with an expected linear convergence rate which can be described by the properties of the matrix A and its column submatrices. The analysis demonstrates that RBGS improves RGS more when given appropriate column-paving of the matrix, a partition of the columns into well-conditioned blocks. The main result yields a RBGS method that is more e cient than the simple RGS method.

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Anderson, Curtis James. "Estimating the Optimal Extrapolation Parameter for Extrapolated Iterative Methods When Solving Sequences of Linear Systems." University of Akron / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=akron1383826559.

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Jurča, Ondřej. "Ustálený chod a zkratové poměry v síti 110 kV E.ON napájené z rozvodny 110 kV Otrokovice v roce 2011." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2011. http://www.nusl.cz/ntk/nusl-219016.

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Distribution network 110 kV owned by E. ON in the area Otrokovice; powered by 110 kV and two variants of involvement contained.The first option is basic involvement, without the use of the bridge. The second option includes involvement with the bridge. The aim of this study is to compare; by calculating the steady-state network operation and short circuit conditions of the network, the involvement of these two options. The thesis is divided into two parts, theoretical and practical. The theoretical part consists of a description of the steady operation of networks of high-voltage and short circuit ratio calculations. Load flow calculations are described by the Gauss-Seidel and Newton iterative method. In the case of short-circuit conditions, the effects of their characteristic values, processes and various methods of calculation are described.In the second part, this theoretical knowledge is applied to input data and dispatching programme with the appropriate calculations of network operation and short circuit conditions. The calculated values are listed in the thesis, on the basis of which an evaluation of the two possible connections is made.

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Book chapters on the topic "Gauss-Seidel iterative method"

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Freundl, Christoph, and Ulrich Rüde. "Gauss-Seidel Iterative Method for the Computation of Physical Problems." In Algorithms Unplugged, 295–304. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-15328-0_30.

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Guo, Peng, and Shi-liang Wu. "A Modified Gauss-Seidel Iteration Method for Solving Absolute Value Equations." In Simulation Tools and Techniques, 137–48. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72792-5_13.

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Najafi, H. Saberi, and S. A. Edalatpanah. "Verification of Iterative Methods for the Linear Complementarity Problem." In Handbook of Research on Modern Optimization Algorithms and Applications in Engineering and Economics, 545–80. IGI Global, 2016. http://dx.doi.org/10.4018/978-1-4666-9644-0.ch021.

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In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.

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"Section 57 The Gauss/Seidel iterative method." In Essentials Engineering Mathematics, 499–505. Chapman and Hall/CRC, 2004. http://dx.doi.org/10.1201/b16977-59.

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"Jacobi and Gauss-Seidel Methods." In Applied Iterative Methods, 113–21. A K Peters/CRC Press, 2007. http://dx.doi.org/10.1201/b10651-15.

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"Jacobi and Gauss-Seidel Methods." In Iterative Optimization in Inverse Problems, 167–78. Chapman and Hall/CRC, 2014. http://dx.doi.org/10.1201/b16485-13.

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Albreem, Mahmoud. "Efficient Iterative Massive MIMO Detectors Based on Iterative Matrix Inversion Methods." In Design Methodologies and Tools for 5G Network Development and Application, 175–95. IGI Global, 2021. http://dx.doi.org/10.4018/978-1-7998-4610-9.ch009.

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Massive multiple-input multiple-output (MIMO) is a key technology in fifth generation (5G) communication systems. Although the maximum likelihood (ML) obtains an optimal performance, it is prohibited in realization because of its high computational complexity. Linear detectors are an alternative solution, but they contain a matrix inversion which is not hardware friendly. Several methods have been proposed to approximate or to avoid the computation of exact matrix inversion. This chapter garners those methods and study their applicability in massive MIMO system so that a generalist in communication systems can differentiate between different algorithms from a wide range of solutions. This chapter presents the performance-complexity profile of a detector based on the Neuamnn-series (NS), Newton iteration (NI), successive over relaxation (SOR), Gauss-Seidel (GS), Jacobi (JA), Richardson (RI), optimized coordinate descent (OCD), and conjugate-gradient (CG) methods in 8×64, 16×64, and 32×64 MIMO sizes, and modulation scheme is 64QAM.

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Conference papers on the topic "Gauss-Seidel iterative method"

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Hhonghao, He, Yuan Dongjin, Hou Yi, and Xu Jinqiu. "Preconditioned Gauss-Seidel Iterative Method for Linear Systems." In 2009 International Forum on Information Technology and Applications (IFITA). IEEE, 2009. http://dx.doi.org/10.1109/ifita.2009.339.

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Uchikawa, Hironori, Brian M. Kurkoski, Kenta Kasai, and Kohichi Sakaniwa. "Iterative encoding with Gauss-Seidel method for spatially-coupled low-density lattice codes." In 2012 IEEE International Symposium on Information Theory - ISIT. IEEE, 2012. http://dx.doi.org/10.1109/isit.2012.6283575.

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Toreja, Allen J., and Rizwan-Uddin. "Improving the Efficiency of the Nodal Integral Method With the Portable, Extensible Toolkit for Scientific Computation." In 10th International Conference on Nuclear Engineering. ASMEDC, 2002. http://dx.doi.org/10.1115/icone10-22684.

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An existing implementation of the nodal integral method for the time-dependent convection-diffusion equation is modified to incorporate various PETSc (Portable, Extensible Toolkit for Scientific Computation) solver and preconditioner routines. In the modified implementation, the default iterative Gauss-Seidel solver is replaced with one of the following PETSc iterative linear solver routines: Generalized Minimal Residuals, Stabilized Biconjugate Gradients, or Transpose-Free Quasi-Minimal Residuals. For each solver, a Jacobi or a Successive Over-Relaxation preconditioner is used. Two sample problems, one with a low Peclet number and one with a high Peclet number, are solved using the new implementation. In all the cases tested, the new implementation with the PETSc solver routines outperforms the original Gauss-Seidel implementation. Moreover, the PETSc Stabilized Biconjugate Gradients routine performs the best on the two sample problems leading to CPU times that are less than half the CPU times of the original implementation.

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Samir, Lemita, Guebbai Hamza, and Khellaf Ammar. "New version of Gauss Seidel iterative method for operators system to approach Fredholm integral equation." In 2019 8th International Conference on Modeling Simulation and Applied Optimization (ICMSAO). IEEE, 2019. http://dx.doi.org/10.1109/icmsao.2019.8880320.

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Mathur, Sanjay R., and Jayathi Y. Murthy. "A Multigrid Method for the Solution of Ion Transport Using the Poisson Nernst Planck Equations." In ASME 2007 InterPACK Conference collocated with the ASME/JSME 2007 Thermal Engineering Heat Transfer Summer Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/ipack2007-33410.

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Recently there has been much interest in simulating ion transport in biological and synthetic ion channels using the Poisson-Nernst-Planck (PNP) equations. However, many published methods exhibit poor convergence rates, particularly at high driving voltages, and for long-aspect ratio channels. The paper addresses the development of a fast and efficient coupled multigrid method for the solution of the PNP equations. An unstructured cell-centered finite volume method is used to discretize the governing equations. An iterative procedure, based on a Newton-Raphson linearization accounting for the non-linear coupling between the Poisson and charge transport equations, is employed. The resulting linear system of equations is solved using an algebraic multigrid method, with coarse level systems being created by agglomerating finer-level equations based on the largest coefficients of the Poisson equation. A block Gauss-Seidel update is used as the relaxation method. The method is shown to perform well for ion transport in a synthetic channel for aspect ratios ranging from 16.67 to 1667 for a range of operating parameters.

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Oeljeklaus, Michael, and H. Günther Natke. "Parallel Interval Algorithm for the Parameter Identification of Linear Elastomechanical Systems." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0709.

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Abstract An interval analytical approach to parameter identification in the frequency domain of mathematical models for linear elasto-mechanical systems is described. A priori information, measurement errors and — if possible — unmeasurable degrees of freedom are modelled in terms of intervals. A parallel iterative update method — based on the interval analytical Gauss-Seidel method — is used to reduce the volume of the parameter search space initially given. The search for the global minimum of the WLS objective function using output residuals is performed on the reduced parameter space in a last step1. Subsystem identification and sub-model synthesis are used in the case of realistic models with a large number of degrees of freedom. Parallelization of the algorithm with respect to subsystems is applied in the case of large structures to reduce the amount of memory and to speed-up the computation. Test results for some simulations for a test structure are given in order to illustrate the method.

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Pao, Y. C., and Erik L. Ritman. "Generalized Algorithms for Interactive Warping Analysis of Porous Cross Sections." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/cie-9070.

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Abstract Algorithms have been developed for warping analysis and calculation of the shearing stresses in a general porous cross section of a long rod when it is subjected to twisting torques at its ends. The shape and dimensions of the cross section full of holes are defined from the binary segmented image data with by a micro-CT scanning technique. Finite difference approximation of the Laplace equation governing the cross-sectional warping leading to the matrix solution by a Gauss-Seidel process is discussed. Method of pointer matrix which keeps the locations of the nonzero elements of the coefficient matrix, is employed to expedite the iterative solution. Computer programs are coded in QuickBASIC language to facilitate plotting of the computed distributions of warping and shearing stresses. The classical torsional problem of square and thin-walled cross sections are used to reexamine the accuracy of the developed algorithms and results are found to be in very good agreement.

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Kumar, Ankan, and Sandip Mazumder. "An Unstructured Reacting Flow Solver With Coupled Implicit Solution of the Species Conservation Equations." In ASME 2008 Heat Transfer Summer Conference collocated with the Fluids Engineering, Energy Sustainability, and 3rd Energy Nanotechnology Conferences. ASMEDC, 2008. http://dx.doi.org/10.1115/ht2008-56145.

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Many reacting flow applications mandate coupled solution of the species conservation equations. A low-memory coupled solver was developed to solve the species transport equations on an unstructured mesh with implicit spatial as well as species-to-species coupling. First, the computational domain was decomposed into sub-domains comprised of geometrically contiguous cells—a process termed internal domain decomposition (IDD). This was done using the binary spatial partitioning (BSP) algorithm. Following this step, for each sub-domain, the discretized equations were developed using the finite-volume method, written in block implicit form, and solved using an iterative solver based on Krylov sub-space iterations, i.e., the Generalized Minimum Residual (GMRES) solver. Overall (outer) iterations were then performed to treat explicitness at sub-domain interfaces and non-linearities in the governing equations. The solver is demonstrated for a laminar ethane-air flame calculation with five species and a single reaction step, and for a catalytic methane-air combustion case with 19 species and 22 reaction steps. It was found that the best performance is manifested for sub-domain size of about 1000 cells, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block Gauss-Seidel procedure.

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Eswaran, M., and U. K. Saha. "Low Steeping Waves Simulation in a Vertical Excited Container Using σ Transformation." In ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2009. http://dx.doi.org/10.1115/omae2009-80248.

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A fluid partially occupying a moving tank undergoes wave motions (sloshing). These motions generate severe hydrodynamic loads that can be dangerous for structural integrity and stability of rockets, satellites, LNG ships, trucks and even stationary petroleum containers. Free surface motions of the liquid in partially filled tanks under gravity are of practical significance particularly in marine and road transportation applications. For this reason, liquid sloshing has always been a research subject attracting great concern during the last several decades. In this paper, a fully non-linear finite difference model has been developed based on the inviscid flow equations, and a simple mapping function was used to remove the time-dependence of the free surface in the fluid domain. The time-varying fluid surface can be mapped onto a rectangular domain by the σ-transformation. This method is a simple way to simulate non-breaking waves quickly and accurately especially that has a low steepness. The fluid motion is solved in a unit square mesh in the transformed flow domain (i.e., computational domain). The fourth order central difference scheme and the Gauss–Seidel point successive over-relaxation iterative procedure are used to capture the free surface wave profiles and the free surface elevation plots of the fluid domain. Difference between the peaks and troughs of waves are discussed for the case of vertical excitation of first three natural frequency of the tank. Phase-plane diagrams are drawn to show the non-linearity of the motion of time dependent free surface. The results agree well with the previously published results.

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Negrut, Dan, Alessandro Tasora, and Mihai Anitescu. "Large-Scale Parallel Multibody Dynamics With Frictional Contact on the GPU." In ASME 2008 Dynamic Systems and Control Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/dscc2008-2139.

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In the context of simulating the frictional contact dynamics of large systems of rigid bodies, this paper reviews a novel method for solving large cone complementarity problems by means of a fixed-point iteration algorithm. The method is an extension of the Gauss-Seidel and Gauss-Jacobi methods with overrelaxation for symmetric convex linear complementarity problems. Convergent under fairly standard assumptions, the method is implemented in a parallel framework by using a single instruction multiple data (SIMD) computation paradigm promoted by the Compute Unified Device Architecture (CUDA) library for graphical processing unit (GPU) programming. The framework is anticipated to become a viable tool for investigating the dynamics of complex systems such as ground vehicles running on sand, powder composites, and granular material flow.

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