An Adaptive Semi-Iterative Method for Symmetric Semidefinite Linear Systems

Abstract

The development of semi-iterative methods for the solution of large linear nonsingular systems has received considerable attention. However, these methods generally cannot be applied to the solution of singular linear systems. We present a new adaptive semi-iterative method tailored for the solution of large sparse symmetric semidefinite linear systems. This method is a modification of Richardson iteration and requires the determination of relaxation parameters. We want to choose relaxation parameters that yield rapid convergence, and this requires knowledge of an interval [a, b] on the real axis that contains most of the nonvanishing eigenvalues of the matrix. Such an interval is determined during the iterations by computing certain modified moments. Computed examples show that our adaptive iterative method typically requires a smaller number of iterations and much fewer inner product evaluations than an appropriate modification of the conjugate gradient algorithm of Hestenes and Stiefel. This makes our scheme particularly attractive to use on certain parallel computers on which the communication required for inner product evaluations constitutes a bottleneck.KeywordsRelaxation ParameterModify MomentConjugate Gradient AlgorithmInitial IntervalRoundoff ErrorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.