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The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s > 0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

The paper is concerned with iterative algorithms for solving systems of linear algebraic equations where the coefficient matrix is large and sparse. Such systems often arise in the numerical solution of partial differential equations by finite difference or finite element methods. The algorithms considered include a basic iterative method, an acceleration procedure for speeding up the convergence of the basic iterative method and an adaptive procedure for determining any necessary iteration parameters. Algorithms are described both for the case where the coefficient matrix is symmetric and positive definite and for the nonsymmetrizable case. The ITPACK Project was established by the Center for Numerical Analysis at The University of Texas in 1974. The object of ITPACK was the study of iterative algorithms, using both theoretical and experimental methods. Several software packages are described which have been developed as part of the ITPACK project for carrying out experimental studies with a variety of algorithms over a wide range of problems. Recent work on the use of advanced computer architectures is briefly discussed. [ABSTRACT FROM AUTHOR]