Comparison Theorems for Splittings of M-matrices in (block) Hessenberg Form
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Some variants of the (block) Gauss-Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix ρ(P_GS)≥ρ(P_S)≥ρ(P_AGS), where P_GS, P_S, P_AGS are the iteration matrices of the Gauss-Seidel, staircase, and anti-Gauss-Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.
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