Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user‐defined region in the complex plane. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solving for multiple eigenpairs simultaneously. The traditional FEAST algorithm is implemented by directly solving collections of shifted linear systems of equations; in this paper, we describe a variation of the FEAST algorithm that uses iterative Krylov subspace algorithms for solving the shifted linear systems inexactly. We show that this iterative FEAST algorithm (which we call IFEAST) is mathematically equivalent to a block Krylov subspace method for solving eigenvalue problems. By using Krylov subspaces indirectly through solving shifted linear systems, rather than directly using them in projecting the eigenvalue problem, it becomes possible to use IFEAST to solve eigenvalue problems using very large dimension Krylov subspaces without ever having to store a basis for those subspaces. IFEAST thus combines the flexibility and power of Krylov methods, requiring only matrix–vector multiplication for solving eigenvalue problems, with the natural parallelism of the traditional FEAST algorithm. We discuss the relationship between IFEAST and more traditional Krylov methods and provide numerical examples illustrating its behavior.     

Compress‐and‐restart block Krylov subspace methods for Sylvester matrix equations

Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well‐explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.     

A compact rational Krylov method for large‐scale rational eigenvalue problems

In this work, we propose a new method, termed as R‐CORK, for the numerical solution of large‐scale rational eigenvalue problems, which is based on a linearization and on a compact decomposition of the rational Krylov subspaces corresponding to this linearization. R‐CORK is an extension of the compact rational Krylov method (CORK) introduced very recently in the literature to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R‐CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R‐CORK is more efficient from the point of view of memory and orthogonalization than the classical rational Krylov method applied directly to the linearization. Taking into account that the R‐CORK method is based on a classical rational Krylov method, to implement implicit restarting is also possible, and we show how to do it in a memory‐efficient way. Finally, some numerical examples are included in order to show that the R‐CORK method performs satisfactorily in practice.     

Rational Krylov sequence methods for eigenvalue computation

Algorithms to solve large sparse eigenvalue problems are considered. A new class of algorithms which is based on rational functions of the matrix is described. The Lanczos method, the Arnoldi method, the spectral transformation Lanczos method, and Rayleigh quotient iteration all are special cases, but there are also new algorithms which correspond to rational functions with several poles. In the simplest case a basis of a rational Krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of Hessenberg matrices.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

Backward error of polynomial eigenvalue problems solved by linearization

It is commonplace in many application domains to utilize polynomial eigenvalue problems to model the behaviour of physical systems. Many techniques exist to compute solutions of these polynomial eigenvalue problems. One of the most frequently used techniques is linearization, in which the polynomial eigenvalue problem is turned into an equivalent linear eigenvalue problem with the same eigenvalues, and with easily recoverable eigenvectors. The eigenvalues and eigenvectors of the linearization are usually computed using a backward stable solver such as the QZ algorithm. Such backward stable algorithms ensure that the computed eigenvalues and eigenvectors of the linearization are exactly those of a nearby linear pencil, where the perturbations are bounded in terms of the machine precision and the norms of the matrices defining the linearization. Although we have solved a nearby linear eigenvalue problem, we are not certain that our computed solution is in fact the exact solution of a nearby polynomial eigenvalue problem. Here, we perform a backward error analysis for the solution of a specific linearization for polynomials expressed in the monomial basis. We use a suitable one-sided factorization of the linearization that allows us to map generic perturbations of the linearization onto structured perturbations of the polynomial coefficients. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems

The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A, but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F-orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F-Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F-Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.     

A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh–Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem.     

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace based on a square matrix sequence {A _j } and a vector sequence {u _j }. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of . By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods. Yimin Wei is supported by the National Natural Science Foundation of China and Shanghai Education Committee.     

On hybrid iterative methods for nonsymmetric systems of linear equations

Summary. Hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired, and a second phase in which a polynomial iteration designed with respect to this information is used. Most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the Arnoldi process in Phase I. We will show the limitations of this approach and propose an alternative, also based on the Arnoldi process, which approximates the field of values of the coefficient matrix and of its inverse in the Krylov subspace. We also report on numerical experiments comparing the resulting new method with other hybrid algorithms. Received May 27, 1993 / Revised version received November 14, 1994     

Constraint preconditioning for nonsymmetric indefinite linear systems

This paper introduces and presents theoretical analyses of constraint preconditioning via a Schilders'‐like factorization for nonsymmetric saddle‐point problems. We extend the Schilders' factorization of a constraint preconditioner to a nonsymmetric matrix by using a different factorization. The eigenvalue and eigenvector distributions of the preconditioned matrix are determined. The choices of the parameter matrices in the extended Schilders' factorization and the implementation of the preconditioning step are discussed. An upper bound on the degree of the minimum polynomial for the preconditioned matrix and the dimension of the corresponding Krylov subspace are determined, as well as the convergence behavior of a Krylov subspace method such as GMRES. Numerical experiments are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

A review of formal orthogonality in Lanczos-based methods

Krylov subspace methods and their variants are presently the favorite iterative methods for solving a system of linear equations. Although it is a purely linear algebra problem, it can be tackled by the theory of formal orthogonal polynomials. This theory helps to understand the origin of the algorithms for the implementation of Krylov subspace methods and, moreover, the use of formal orthogonal polynomials brings a major simplification in the treatment of some numerical problems related to these algorithms. This paper reviews this approach in the case of Lanczos method and its variants, the novelty being the introduction of a preconditioner.     

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter‐dependent matrix. In several applications, M(λ) has a structure where the higher‐order terms of its Taylor expansion have a particular low‐rank structure. We propose a new Arnoldi‐based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector‐valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper, the low‐rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large‐scale problems. Copyright © 2016 John Wiley & Sons, Ltd.     

An Arnoldi based algorithm for large algebraic Riccati equations

In the present work, we present a numerical method for the computation of approximate solutions for large continuous-time algebraic Riccati equations. The proposed method is a method of projection onto a matrix Krylov subspace. We use a matrix Arnoldi process to construct an orthonormal basis. We give some theoretical results and numerical experiments for large problems.     

A computational method for large‐scale differential symmetric Stein equation

We propose a numerical method for solving large‐scale differential symmetric Stein equations having low‐rank right constant term. Our approach is based on projection the given problem onto a Krylov subspace then solving the low dimensional matrix problem by using an integration method, and the original problem solution is built by using obtained low‐rank approximate solution. Using the extended block Arnoldi process and backward differentiation formula (BDF), we give statements of the approximate solution and corresponding residual. Some numerical results are given to show the efficiency of the proposed method.     

Inverse subspace problems with applications

Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large‐scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace‐restricted SVD method for the solution of linear discrete ill‐posed problems, also are considered.Copyright © 2013 John Wiley & Sons, Ltd.     

Improving projection‐based eigensolvers via adaptive techniques

We consider subspace iteration (or projection‐based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near‐to‐optimum overall work without detailed a priori knowledge about the eigenvalue distribution. The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E. Polizzi: Density‐matrix‐based algorithm for solving eigenvalue problems. Phys. Rev. B 2009; 79 :115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.     

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.     

Block Krylov Subspace Methods for Solving Large Sylvester Equations

In the present paper, we propose block Krylov subspace methods for solving the Sylvester matrix equation AX–XB=C. We first consider the case when A is large and B is of small size. We use block Krylov subspace methods such as the block Arnoldi and the block Lanczos algorithms to compute approximations to the solution of the Sylvester matrix equation. When both matrices are large and the right-hand side matrix is of small rank, we will show how to extract low-rank approximations. We give some theoretical results such as perturbation results and bounds of the norm of the error. Numerical experiments will also be given to show the effectiveness of these block methods.     

Subspace methods for three‐parameter eigenvalue problems

We propose subspace methods for three‐parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for two‐parameter eigenvalue problems exist, their extensions to a three‐parameter setting seem challenging. An inherent difficulty is that, while for two‐parameter eigenvalue problems, we can exploit a relation to Sylvester equations to obtain a fast Arnoldi‐type method, such a relation does not seem to exist when there are three or more parameters. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi–Davidson‐type method for three or more parameters, which we generalize from its two‐parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi–Davidson approach is devised to locate eigenvalues close to a prescribed target; yet, it often also performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. The proposed approaches are suitable especially for problems where the computation of several eigenvalues is required with high accuracy. MATLAB implementations of both methods have been made available in the package MultiParEig (see http://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig ).