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This is a final report on work performed under NASA grant NAG-1-1112-FOP during the period March, 1990 through February 1995. Four major topics are covered: (1) solution of nonlinear poisson-type equations; (2) parallel reduced system conjugate gradient method; (3) orderings for conjugate gradient preconditioners, and (4) SOR as a preconditioner.

This report summarizes work on parallel computations for NASA Grant NAG-1-1529 for the period 1 Jan. - 30 June 1994. Short summaries on highly parallel preconditioners, target-specific parallel reductions, and simulation of delta-cache protocols are provided.

The Arnoldi (1951) and Lanczos (1950) iterative algorithms are used to obtain a few selected eigenvalues of the generalized eigenvalue problem A(x) = lambdaB(x), where A and B are n x n nonsymmetric banded complex matrices. A shift-and-invert strategy is used to increase the rate of convergence toward the desired eigenvalues, and the two approaches are evaluated in view of their performance in a model problem concerning the linear stability of compressible boundary layers. A general scheme for computing the eigenvalues that lie within a 'box' in the complex plane is presented.

The effect of orderings on the rate of convergence of the conjugate gradient method with SSOR or incomplete Cholesky preconditioning is examined. Some results also are presented that help to explain why red/black ordering gives an inferior rate of convergence.