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Theorems of Stein-Rosenberg type. III. The singular case
Authors:John J. Buoni  Michael Neumann  Richard S. Varga
Affiliation:Youngstown State University Youngstown, Ohio 44555 U.S.A.;University of South Carolina Columbia, South Carolina 29208 U.S.A.;Kent State University Kent, Ohio 44242 U.S.A.
Abstract:In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix Jω and the successive overrelaxation (SOR) matrix
/></figure>, in the case when the Jacobi matrix <em>J</em><sub>1</sub> is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of <em>J</em><sub>ω</sub> and <figure class=/></figure> have been established which do not depend on the assumption that <em>J</em><sub>1</sub> is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix <em>A</em> with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that <em>J</em><sub>ω</sub> and <figure class=/></figure> simultaneously semiconverge for all ω in the real interval [0,1).</td>
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