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Comparative Study on Solving Methods of Linear Equations of Sparse Matrix – Take The Direct Method and Gmres for Example

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Rong, Y., 2021. Comparative Study on Solving Methods of Linear Equations of Sparse Matrix – Take The Direct Method and Gmres for Example. United International Journal for Research & Technology (UIJRT), 2(11), pp.91-97.


With the increase of the complexity of practical problems, the optimization of sparse linear equations becomes more and more important. Based on two kinds of solutions of linear equations: direct solution and iterative solution, this paper probes into the solutions of sparse matrix equations. For the direct solution, the Gauss elimination method without selecting pivot elements, column selecting pivot elements, and all selecting pivot elements is selected to explore the performance of sparse matrix solution for triplet storage. For the iterative method, we select the GMRES method and explore three methods including general GMRES, GMRES with LU pretreatment, and GMRES with LU pretreatment and restart. Finally, we compare and analyze the performance differences of all the above direct and iterative solutions, and find that the GMRES method restarted after incomplete LU decomposition can be used as a better general solution of general linear sparse matrix equations. At the same time, the applicable matrices of various methods and their shortcomings are given.

Keywords: Sparse matrix, GMRES, Restart, LU pretreatment, Optimization, direct solution.


  1. Saad, M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Society for Industrial and Applied Mathematics, 1986, 7(3): 856-869.
  2. Ma xiaofei.(2013). Review of research and application of generalized Minimum residual method. Pure Mathematics, 03(3),181-187.
  3. Yang Z. Research on GMRES method and its variation Algorithm. (Dissertation, University of Electronic Science and Technology of China).
  4. Arnoldi W E. The principle of minimized iteration in the solution of the matrix eigenproblem[C]// Quart. Appl. Math. 1951:17-29
  5. Gao HAIyan. (2008). Comparison of SOR and GMRES methods for solving linear equations. Science and Technology Information: Academic Edition, 000(036), 106-107.

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