A comparison theorem for the iterative method with the preconditioner (I + Smax)
Journal of Computational and Applied Mathematics
On optimal improvements of classical iterative schemes for Z-matrices
Journal of Computational and Applied Mathematics
A note on the preconditioner Pm=(I+Sm)
Journal of Computational and Applied Mathematics
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Kotakemori et al. (2002) [2] have reported that the convergence rate of the iterative method with a preconditioner P"m=(I+S"m"a"x) was superior to one of the modified Gauss-Seidel methods under a special condition. The authors derived a theorem comparing the Gauss-Seidel method. To remove the requirement for this condition, Morimoto et al. (2004) [4] have proposed the preconditioner P"s"m=(I+S+S"m). However, it is pointed out that there exists a special matrix that does not satisfy this comparison theorem. To overcome this problem, Kohno et al. (2009) [3] have proposed some preconditioners. In this note, we present a new preconditioner and from numerical results, we show that the convergence rate of the proposed method is better than that of the Gauss-Seidel method with other preconditioners. In addition, we presented the comparison theorem for the proposed preconditioner. We succeeded to overcome two drawbacks mentioned above.