Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices

The progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a different short-term recurrence method based on Krylov subspaces for such matrices, which can be used as either a solver or a preconditioner. Numerical experiments compare this method to alternative algorithms.

[1]  R. Cottle.  Manifestations of the Schur complement , 1974 .

[2]  M. Rozložník,et al.  Numerical behaviour of the modified gram-schmidt GMRES implementation , 1997 .

[3]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[4]  Miroslav Rozlozník,et al.  Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES , 2006, SIAM J. Matrix Anal. Appl..

[5]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[6]  Gene H. Golub,et al.  A generalized conjugate gradient method for non-symmetric systems of linear equations , 2007, Milestones in Matrix Computation.

[7]  Marko Huhtanen,et al.  A Matrix Nearness Problem Related to Iterative Methods , 2001, SIAM J. Numer. Anal..

[8]  E. Yip.  A Note on the Stability of Solving a Rank-p Modification of a Linear System by the Sherman–Morrison–Woodbury Formula , 1986 .

[9]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[10]  Valeria Simoncini,et al.  On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods , 2005, SIAM Rev..

[11]  Y. Saad,et al.  On the Lánczos method for solving symmetric linear systems with several right-hand sides , 1987 .

[12]  Josef Sifuentes,et al.  Preconditioned iterative methods for inhomogeneous acoustic scattering applications , 2010 .

[13]  Vladimir Rokhlin,et al.  On the inverse scattering problem for the Helmholtz equation in one dimension , 1992 .

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

[15]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[16]  Martin B. van Gijzen,et al.  IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations , 2008, SIAM J. Sci. Comput..

[17]  J. Sherman,et al.  Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix , 1950 .

[18]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[19]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[20]  B. Fischer.  Polynomial Based Iteration Methods for Symmetric Linear Systems , 1996 .

[21]  V. Simoncini,et al.  Convergence properties of block GMRES and matrix polynomials , 1996 .

[22]  Lothar Reichel,et al.  The Arnoldi Process and GMRES for Nearly Symmetric Matrices , 2008, SIAM J. Matrix Anal. Appl..

[23]  William W. Hager,et al.  Updating the Inverse of a Matrix , 1989, SIAM Rev..

[24]  Josef A. Sifuentes,et al.  THE STABILITY OF GMRES CONVERGENCE, WITH APPLICATION TO APPROXIMATE DEFLATION PRECONDITIONING , 2011 .

[25]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[26]  O. Widlund.  A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations , 1978 .

[27]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[28]  D. O’Leary.  The block conjugate gradient algorithm and related methods , 1980 .

[29]  M. Gutknecht.  BLOCK KRYLOV SPACE METHODS FOR LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES : AN , 2005 .

[30]  Christopher C. Paige,et al.  A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm n-Vectors , 2009, SIAM J. Matrix Anal. Appl..

[31]  Raf Vandebril,et al.  An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices , 2010 .

[32]  Tobin A. Driscoll,et al.  From Potential Theory to Matrix Iterations in Six Steps , 1998, SIAM Rev..

[33]  Valeria Simoncini,et al.  Interpreting IDR as a Petrov--Galerkin Method , 2010, SIAM J. Sci. Comput..

[34]  Claude Brezinski.  Schur Complements and Applications in Numerical Analysis , 2005 .

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