In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution

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... of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.