A Restarted GMRES Method Augmented with Eigenvectors

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.

[1]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[2]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[3]  J. G. Lewis,et al.  Recent research in iterative methods at Boeing , 1989 .

[4]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[5]  Lloyd N. Trefethen,et al.  How Fast are Nonsymmetric Matrix Iterations? , 1992, SIAM J. Matrix Anal. Appl..

[6]  S. Ashby.  Polynomial Preconditioning for Conjugate Gradient Methods , 1988 .

[7]  Martin H. Gutknecht,et al.  A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part I , 1992, SIAM J. Matrix Anal. Appl..

[8]  H. Elman.  Iterative methods for large, sparse, nonsymmetric systems of linear equations , 1982 .

[9]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[10]  H. V. D. Vorst,et al.  The rate of convergence of Conjugate Gradients , 1986 .

[11]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[12]  M. Gutknecht.  A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms. Part II , 1994, SIAM J. Matrix Anal. Appl..

[13]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[14]  A. K. Cline.  Several Observations on the Use of Conjugate Gradient Methods , 1978 .

[15]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[16]  O. Axelsson,et al.  On the rate of convergence of the preconditioned conjugate gradient method , 1986 .

[17]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[18]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[19]  A. Bruaset.  A survey of preconditioned iterative methods , 1995 .

[20]  Wayne Joubert,et al.  Lanczos Methods for the Solution of Nonsymmetric Systems of Linear Equations , 1992, SIAM J. Matrix Anal. Appl..

[21]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[22]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[23]  R. Morgan.  Computing Interior Eigenvalues of Large Matrices , 1991 .

[24]  Y. Saad.  Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices , 1980 .

[25]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[26]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .

[27]  Y. Saad.  Krylov subspace methods for solving large unsymmetric linear systems , 1981 .

[28]  Y. Saad,et al.  A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations , 1986 .