Weighted Matrix Ordering and Parallel Banded Preconditioners for Iterative Linear System Solvers
SIAM Journal on Scientific Computing
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We prove the following theorem, which includes a conjecture by Greenbaum and Rodrigue [[ BIT, 29 (1989), pp. 610--634] as a special case: Let Ah be the five-point Laplace matrix on a unit square for grid size h, and let Mh be any symmetric positive definite preconditioner for Ah with half bandwidth k=O(1). Then the condition number $\kappa(M_h^{-{1\over 2}}A_hM_h^{-{1\over 2}})$ satisfies $\kappa(M_h^{-{1\over 2}}A_hM_h^{-{1\over 2}}) = O(h^{-2}).