Summary
Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the ‘usual’D−L−U, and (ii) of an hermitian positive definite matrixA intoD−L−U, whereD is hermitian positive definite andL=1/2(A−D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in ω, ω≧0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].
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Research supported in part by the Air Force Office of Scientific Research, and the Department of Energy
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Neumann, M., Varga, R.S. On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices. Numer. Math. 35, 69–79 (1980). https://doi.org/10.1007/BF01396371
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DOI: https://doi.org/10.1007/BF01396371