Study on the preconditioners (I+Sm)(I+Sm)

Kotakemori et al. (2002) [2] have reported that the convergence rate of the iterative method with a preconditioner Pm=(I+Smax)Pm=(I+Smax) 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 Psm=(I+S+Sm)Psm=(I+S+Sm). 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.