Note on stationary iterative methods by SVD

摘要

Golub and Pillis's [G. Golub, J. de Pillis, Toward an effective two-parameter SOR method, in: D.R. Kincaid, L. Hayes (Eds.), Iterative methods for Large Linear Systems, Academic Press, New York, 1990, pp. 107–118] have given a good proof of Young's [D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971] eigenvalue functional of the SOR method for block symmetric matrices with Property A by the singular value decomposition. Here we use the same way to treat the AOR method proposed by Hadjidimos [A. Hadjidimos, Math. Comput. 32 (1978) 149–157] for block symmetric matrices with Property A. We also study the SOR method, the AOR method and the two-parameter SOR method for block skew-symmetric matrices by singular value decomposition (SVD). For skew-symmetric matrices, we just re-derive the eigenvalue functional for the three methods, but obtain the different results for the two-norm of the iterative matrices.