GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Iterative methods for solving linear systems
Iterative methods for solving linear systems
On the Construction of Deflation-Based Preconditioners
SIAM Journal on Scientific Computing
GMRES with Deflated Restarting
SIAM Journal on Scientific Computing
Implicitly Restarted Arnoldi Methods and Subspace Iteration
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing
SIAM Journal on Scientific Computing
Inexact Krylov Subspace Methods for Linear Systems
SIAM Journal on Matrix Analysis and Applications
Variable Accuracy of Matrix-Vector Products in Projection Methods for Eigencomputation
SIAM Journal on Numerical Analysis
A Comparison of Deflation and the Balancing Preconditioner
SIAM Journal on Scientific Computing
Inexact Inverse Iteration with Variable Shift for Nonsymmetric Generalized Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices
SIAM Journal on Matrix Analysis and Applications
Inexact Inverse Subspace Iteration with Preconditioning Applied to Non-Hermitian Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
SIAM Journal on Matrix Analysis and Applications
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We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155-1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.