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Convergence for nonnegative double splittings of matrices. (English) Zbl 1232.65056
The authors consider $n$ by $n$ systems of linear equations with regular matrix $A$ and double splittings $A=P-R-S$ leading to 2-step iterations $P{x}^{i+1}=R{x}^{i}+S{x}^{i-1}$. They prove convergence theorems like “a nonnegative double splitting is convergent iff the simple splitting ($A=P-\left(R+S\right)$) is” and comparison theorems for different double splittings of the same or of different matrices A. Explicit examples concern 2 by 2 matrices, there is no result on an advantage of double over simple splittings.
##### MSC:
 65F10 Iterative methods for linear systems
##### References:
 [1] Avdelas, G., De Pillis, J., Hadjidimos, A., Neumann, M.: A guide to the acceleration of iterative methods whose iteration matrix is nonnegative and convergent. SIAM J. Matrix Anal. Appl. 9, 329–342 (1988) · Zbl 0657.65051 · doi:10.1137/0609027 [2] Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematics Sciences. Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia (1994) [3] Bru, R., Ginestar, D., Marín, J., Verdú, G., Mas, J., Manteuffel, T.: Iterative schemes for the neutron diffusion equation. Comput. Math. Appl. 44, 1307–1323 (2002) · Zbl 1103.82321 · doi:10.1016/S0898-1221(02)00258-4 [4] Cvetković, L.J.: Two-sweep iterative methods. Nonlinear Anal. Theory Methods Appl. 30, 25–30 (1997) · Zbl 0889.65025 · doi:10.1016/S0362-546X(97)00002-3 [5] Marek, I., Szyld, D.B.: Comparison theorems for weak splittings of bounded operators. Numer. Math. 58, 387–397 (1990) · Zbl 0694.65023 · doi:10.1007/BF01385632 [6] Miao, S., Zheng, B.: A note on double splittings of different monotone matrices. Calcolo 46, 261–266 (2009) · Zbl 1185.65058 · doi:10.1007/s10092-009-0011-z [7] Shen, S., Huang, T.: Convergence and comparison theorems for double splittings of matrices. Comput. Math. Appl. 51, 1751–1760 (2006) · Zbl 1134.65341 · doi:10.1016/j.camwa.2006.02.006 [8] Shen, S., Huang, T., Shao, J.: Convergence and comparison results for double splittings of Hermitian positive definite matrices. Calcolo 44, 127–135 (2007) · Zbl 1150.65008 · doi:10.1007/s10092-007-0132-1 [9] Song, Y.: Comparisons of nonnegative splittings of matrices. Linear Algebra Appl. 154–156, 433–455 (1991) · Zbl 0732.65024 · doi:10.1016/0024-3795(91)90388-D [10] Song, Y.: Comparison theorems for nonnegative splittings of bounded operators. Appl. Math. 12, 137–142 (1999) [11] Song, Y.: Comparison theorems for splittings of matrices. Numer. Math. 92, 563–591 (2002) · Zbl 1012.65028 · doi:10.1007/s002110100333 [12] Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics, vol. 27. Springer, Berlin (2000) [13] Woźnicki, Z.I.: Estimation of the optimum relaxation factors in partial factorization iterative methods. SIAM J. Matrix. Anal. Appl. 14, 59–73 (1993) · Zbl 0767.65025 · doi:10.1137/0614005 [14] Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971) [15] Zhang, C.: On convergence of double splitting methods for non-Hermitian positive semidefinite linear systems. Calcolo 47, 103–112 (2010) · Zbl 1204.65033 · doi:10.1007/s10092-009-0015-8