We study the use of Krylov subspace recycling for the solution of a sequence of slowlychanging families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework.As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.

Abstract. We present a detailed convergence analysis of preconditioned MINRES for approximately solving the linear systems that arise when Rayleigh Quotient Iteration is used to compute the lowest eigenpair of a symmetric positive definite matrix. We provide insight into the "slow start" of MINRES iteration in both a qualitative and quantitative way, and show that the convergence of MINRES mainly depends on how quickly the unique negative eigenvalue of the preconditioned shifted coefficient matrix is approximated by its corresponding harmonic Ritz value. By exploring when the negative Ritz value appears in MINRES iteration, we obtain a better understanding of the limitation of preconditioned MINRES in this context and the virtue of a new type of preconditioner with "tuning". Comparison of MINRES with SYMMLQ in this context is also given. Finally we show that tuning based on a rank-2 modification can be applied with little additional cost to guarantee positive definiteness of the tuned preconditioner.Key words. Rayleigh Quotient Iteration, harmonic Ritz value, MINRES, tuned preconditioner 1. Introduction. There has been considerable interest in recent years in developing and analyzing eigensolvers with inner-outer structure for computing eigenvalues of matrices closest to some specified value. These algorithms usually involve at each step (outer iteration) a shift-invert matrix-vector product implemented by solving the shifted linear system iteratively (inner iteration). The use of inner iteration becomes mandatory if the matrices are too large for factorization-based exact shift-invert matrix-vector products to be practical. Inexact inverse iteration is the most simple algorithm of this type and the best understood one. Early papers on the convergence of inexact inverse iteration with fixed shift include [8] and [9], where the main concern is to choose a decreasing sequence of stopping tolerances for inner solvers to maintain linear convergence of the outer iteration. Analysis of inexact Rayleigh Quotient Iteration (RQI) in [13] and [18] shows how the inexactness of the inner solve can affect the convergence of the outer iteration. More recent work focuses on improving the convergence of inner iterations as well as the relation between the inner and outer iterations. Reference [16] introduces some new perspectives on preconditioning in this setting, namely, that faster convergence of inner iterations can be obtained by modifying the right hand side of the preconditioned linear system. Refined analysis of this approach in [1], [2] and [5] shows how different formulations of the linear system, with variable shift and different inner stopping criteria, can affect the convergence of the inner and outer iterations. An alternative preconditioning approach called "tuning" is analyzed in [6] for non-symmetric eigenvalue problems and in [7] for symmetric problems. A preconditioner with tuning is a low rank modification of an ordinary preconditioner. Tuning forces the preconditioning operator to behave in the same way as...

Abstract. We introduce the Generalized Preconditioned Locally Harmonic Residual (GPLHR) method for solving standard and generalized non-Hermitian eigenproblems. The method is particularly useful for computing a subset of eigenvalues, and their eigen-or Schur vectors, closest to a given shift. The proposed method is based on block iterations and can take advantage of a preconditioner if it is available. It does not need to perform exact shift-and-invert transformation. Standard and generalized eigenproblems are handled in a unified framework. Our numerical experiments demonstrate that GPLHR is generally more robust and efficient than existing methods, especially if the available memory is limited.

Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T ( λ ) v = 0 T(\lambda )v=0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, including its global and asymptotic convergence. We propose several variants of single-vector and block PCG methods with deflation for computing multiple eigenvalues, and compare them in arithmetic and memory cost. Variable indefinite preconditioning is shown to be effective to accelerate convergence when some desired eigenvalues are not close to the lowest or highest eigenvalue. The efficiency of variants of PCG is illustrated by numerical experiments. Overall, the locally optimal block preconditioned conjugate gradient (LOBPCG) is the most efficient method, as in the linear setting.

We study inexact subspace iteration for solving generalized non-Hermitian eigenvalue problems with spectral transformation, with focus on a few strategies that help accelerate preconditioned iterative solution of the linear systems of equations arising in this context. We provide new insights into a special type of preconditioner with "tuning" that has been studied for this algorithm applied to standard eigenvalue problems. Specifically, we propose an alternative way to use the tuned preconditioner to achieve similar performance for generalized problems, and we show that these performance improvements can also be obtained by solving an inexpensive least squares problem. In addition, we show that the cost of iterative solution of the linear systems can be further reduced by using deflation of converged Schur vectors, special starting vectors constructed from previously solved linear systems, and iterative linear solvers with subspace recycling. The effectiveness of these techniques is demonstrated by numerical experiments.