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THE INCOMPLETE FACTORIZATION METHOD FOR ITERATIVELY SOLVING SYSTEMS OF DIFFERENCE EQUATIONS AND ADAPTATION OF THIS METHOD TO NON-M-MATRICES

V. P. Ginkin, K. G. Chernov, Yu. G. Bartenev, Yu. A. Bondarenko, R. M. Shagaliev, E. B. Shchanikova
VANT. Ser.: Mat. Mod. Fiz. Proc 2009. Вып.3. С. 3-17.

New effective preconditioners - DIF and PIF - of the stabilized method of biadjoint gradients have been developed to solve systems of 2D and 3D elliptic finite-difference equations with asymmetric poorly conditioned M-matrices of coefficients. To solve problems with non-M-matrices, we offer a stable and effective method of eliminating positive non-diagonal elements from a preconditioned matrix with simultaneously increasing its diagonal elements by the summarized value of all the elements eliminated from a given matrix row.
The rate of convergence of the methods has been tested using both Dirichlet and von Neumann test problems for diffusion and diffusion- convection equations and real 9- and 27-diagonal positively defined non-M-matrices.

Keywords: systems of linear equations, iterative methods, method of conjugate gradients, BICGSTAB method, preconditioner, incomplete factorization method, two-dimensional problems, three-dimensional problems, 9-diagonal matrices, 27-diagonal matrices, nonsymmetric matrices, non-M-matrices, paralleling.








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