A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon's thick restarting approach [SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602-616]. It has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns. The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES. Also, it is demonstrated that using harmonic Ritz vectors is important because then the whole subspace is a Krylov subspace that contains certain important smaller subspaces. [38,35] for solving large systems of linear equations. It deflates eigenvalues using the thick restarting technique given by Wu and Simon in [48] for the Lanczos eigenvalue method. Deflation can significantly improve the convergence of restarted GMRES, and it helps robustness by allowing the solution of many tough problems that have small eigenvalues. Introduction. We develop a new version of GMRESThe new method, called GMRES with deflated restarting or GMRES-DR, is mathematically equivalent to the method GMRES with eigenvectors [23] and to two methods in [25], including implicitly restarted GMRES. Thus this paper could be titled, "Yet another equivalent deflated GMRES method." However, the new approach has the efficiency of implicit restarting but is simpler and does not have the numerical concerns. Thus it should be a useful improvement.In the next section, we discuss deflation for Krylov methods, including some previous approaches and results. Section 3 presents the new method, including a variant that removes approximate eigenvectors from the Krylov subspace. Then, in section 4, numerical examples are given that illustrate some points, including the importance of having Krylov subspaces with approximate eigenvectors as starting vectors.

The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that approximate eigenvectors corresponding to a few of the smallest eigenvalues be formed and added to the subspace for GMRES. The convergence can be much faster, and the minimum residual property is retained.

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This paper analyzes Davidson's method for computing a few eigenpairs of large sparse symmetric matrices. An explanation is given for why Davidson's method often performs well but occasionally performs very badly. Davidson's method is then generalized to a method which offers a powerful way of applying preconditioning techniques developed for solving systems of linear equations to solving eigenvalue problems. AMS(MOS) subject classifications. 65, 15 2. Davidson's method. Davidson [2] introduced a new method for computing a few eigenvalues of sparse symmetric matrices arising in quantum chemistry calculations.The standard solution technique for such problems is the Lanczos algorithm [7, Chap. 13] which is a clever implementation of the Rayleigh-Ritz procedure applied to a Krylov subspace (that is, a space of the form span (s, As,..., Aks)). Davidson's method also uses the Rayleigh-Ritz procedure (see [7, p. 213]) but on a non-Krylov subspace. Formally Davidson's method is as follows.

Increased understanding of the content of participant reactions to training is a necessary step in improving their construct validity and usefulness. This study examines the factor structure of a large database of participant reactions to training and explores the emergence of a utility factor. The results suggest that participant reactions are multidimensional and that utility judgments represent an underlying dimension. Treating reactions as unidimensional may mask their true relationship to other measures of training effectiveness. © 2000 by Jossey‐Bass, A Publishing Unit of John Wiley & Sons, Inc.