Abstract
In this paper, monotonicity of iterative methods for solving general solvable singularly systems is discussed. The monotonicity results given by Berman, Plemmons, and Semal are generalized to singular systems. It is shown that for an iterative method introduced by a nonnegative splitting of the coefficient matrix there exist some initial guesses such that the iterative sequence converges towards a solution of the system from below or from above. The monotonicity of the block Gauss-Seidel method for solving a p-cyclic system and Markov chain is considered.
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REFERENCES
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
P. J. Courtois and P. Semal, Block iterative algorithms stochastic matrices, Linear Algebra Appl., 76 (1986), pp. 59-70.
A. Hadjidimos, On the optimization of the classical iterative schemes for the solution of complex singular linear systems, SIAM J. Alg. Disc. Meth., 6 (1985), pp. 555-566.
A. Hadjidimos and R. J. Plemmons, Optimal p-cyclic SOR, Numer. Math., 67 (1994), pp. 475-490.
K. Kontovasilis, R. J. Plemmons, and W. J. Stewart, Block cyclic SOR for Markov chains with p-cyclic infinitesimal generator, Linear Algebra Appl., 154-156 (1991), pp. 145-223.
C. D. Meyer, Jr. and R. J. Plemmons, Convergent powers of a matrix with applications to iterative methods for singular linear systems, SIAM J. Numer. Anal., 14 (1977), pp. 699-705.
M. Neumann and R. J. Plemmons, Convergent nonnegative matrices and iterative methods for consistent linear systems, Numer. Math., 31 (1978), pp. 265-279.
R. J. Plemmons, Regular splittings and the discrete Neumann problem, Numer. Math., 25 (1976), pp. 153-161.
D. J. Rose, Convergent regular splittings for singular M-matrices, SIAMJ. Alg. Disc. Meth., 5 (1984), pp. 133-144.
H. Schneider, Theorems on M-splittings of a singular M-matrix which depend on graph structure, Linear Algebra Appl., 58 (1984), pp. 407-424.
P. Semal, Monotone iterative methods for Markov chains, Linear Algebra Appl., 230 (1995), pp. 35-46.
Y. Song, Semiconvergence of nonnegative splittings for singular matrices, Numer. Math., 85 (2000), pp. 109-127.
Y. Song, Semiconvergence of block SOR method for singular linear systems with pcyclic matrices, J. Comput. Appl. Math., 130 (2001), pp. 217-229.
R. S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics, Vol. 27, Springer-Verlag, Berlin, 2000.
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
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Song, Y. Monotone Convergence of Iterative Methods for Singular Linear Systems. BIT Numerical Mathematics 42, 611–624 (2002). https://doi.org/10.1023/A:1022053915597
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DOI: https://doi.org/10.1023/A:1022053915597