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In a recent work of ours we have solved the problem of the minimization of the spectral radius of the iteration matrix of a p-cyclic successive overrelaxation (SOR) method for the solution of the linear system Ax = b, when the matrix A is block p-cyclic consistently ordered, for what is known as the "one-point" problem, for any p greater than or equal to 3. Particular cases of the "one-point" problem were solved by Young, Varga, Kjellberg, Kredell, Russell and others. In the present work we develop a theory using the results of our previous one and solve first the "two-point" problem special cases of which were solved by Wrigley, Eiermann, Niethammer, Ruttan, Noutsos and others. Secondly, we generalize and extend our theory to cover the "many-point" problem and develop a Young-Eidson's type algorithm for its solution. As possible application areas we mention among others the best block p-cyclic repartitioning for the SOR method and the solution of large scale systems arising in queueing network problems in Markov analysis. (C) 1999 Elsevier Science Inc. All rights reserved. Linear Algebra and Its Applications

Linear systems whose associated block Jacobi iteration matrix B is weakly cyclic generated by the cyclic permutation sigma = (sigma(1), sigma(2),...,sigma(p)) in the spirit of Li and Varga are considered. Regions of convergence for the corresponding block p-cyclic SOR method are derived and the exact convergence domains for real spectra, sigma(B-p), Of the same sign are obtained. Moreover, analytical expressions for two special cases for p = 5 are given and numerical results are presented confirming the theory developed. The tools used for this work are mainly from complex analysis and extensive use of (asteroidal) hypocycloids in the complex plane is made to produce our results. Bit Numerical Mathematics

Matrix Anal. Appl.