Matrix analysis
Iterative solution methods
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The iterative solution ofthe system of linear algebraic equations Ax = b with a nonsingular M-matrix A is considered. A one-step iterative method is constructed which is based on the special weak regular splitting ofthe matrix A. We prove that the obtained iterative method is not only convergent but it has also some further advantageous properties: the maximal rate of convergence, the efficiency from the point of view of computational costs and the qualitative adequacy. We also examine the relation between this splitting and the regular splittings. Finally we construct two-sided monotone sequences to the solution of the above system. These sequences are produced by the iteration based on the weak regular splitting of A, with different suitable starting vectors. The method of the possible determination of these vectors are also indicated.