The past few years have seen a significant increase in research into numerical methods for computing selected eigenvalues of large sparse nonsymmetric matrices. This research has begun to lead to the development of high-quality mathematical software. The software includes codes that implement subspace iteration methods, Arnoldi-based algorithms, and nonsymmetric Lanczos methods. The aim of the current study is to evaluate this state-of-the-art software. In this study we consider subspace iteration and Arnoldi codes. We look at the key features of the codes and their ease of use. Then, using a wide range of test problems, we compare the performance of the codes in terms of storage requirements, execution times, accuracy, and reliability. We also consider their suitability for solving large-scale industrial problems. Based on

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.

This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real non-symmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution of large-scale eigenvalue problems. The paper explains and discusses the use of Chebyshev polynomials and the shift-invert and Cayley ^ transforms as matrix transformations for problems that arise from the discretization df partial differential equations. A few other techniques are described. The reliability of iterative methods is also dealt with by introducing the concept of domain of confidence or trust region. This overview gives the reader an idea of the benefits and the drawbacks of several transformation techniques. We also briefly discuss the current software

Eigenvalues and eigenfunctions of linear operators are important to many areas of ap-plied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

Abstract. Krylov subspace based moment matching algorithms, such as PVL (Padé approximation Via the Lanczos process), have emerged as popular tools for efficient analyses of the impulse response in a large linear circuit. In this work, a new derivation of the PVL algorithm is presented from the matrix point of view. This approach simplifies the mathematical theory and derivation of the algorithm. Moreover, an explicit formulation of the approximation error of the PVL algorithm is given. With this error expression, one may implement the PVL algorithm that adaptively determines the number of Lanczos steps required to satisfy a prescribed error tolerance. A number of implementation issues of the PVL algorithm and its error estimation are also addressed in this paper. A generalization to a multiple-input-multiple-output circuit system via a block Lanczos process is also given.

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. In this work we consider the simultaneous solution of large linear systems of the form Ax (j) = b (j) ; j = 1; : : : ; K where A is sparse and non-Hermitian. We describe single-seed and block-seed projection approaches to these multiple right-hand side problems that are based on the QMR and block QMR algorithms, respectively. We use (block) QMR to solve the (block) seed system and generate the relevant biorthogonal subspaces. Approximate solutions to the non-seed systems are simultaneously generated by minimizing their appropriately projected (block) residuals. After the initial (block) seed has converged, the process is repeated by choosing a new (block) seed from among the remaining non-converged systems and using the previously generated approximate solutions as initial guesses for the new seed and non-seed systems. We give theory for the single-seed case that helps explain the convergence behavior under certain conditions. Implementation details for both the single-seed and b...

. A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A 0 , the eigenpairs of A(t) j (1 \Gamma t)A 0 + tA 1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A 1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: ffl bifurcation ffl ill-conditioning due to non-orthogonal eigenvectors ffl jumping of eigenpaths These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented. Key words. eigenvalues, homo...

. We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify the available choices of methods and catalogue available software. Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, -matrix, matrix polynomial, second-order differential equation, overdamped system, gyroscopic system, linearization, backward error, pseudospectrum, condition number, Krylov methods, Arnoldi method, Lanczos method, Jacobi-Davidson method AMS subject classifications. 65F30 Contents 1 Introduction 2 2 Applications of QEPs 4 2.1 Second-order differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Vibration analysis of structural systems -- Modal superpositio...

This paper sketches the main research developments in the area of computational methods for eigenvalue problems during the 20th century. The earliest of such methods dates back to work of Jacobi in the middle of the nineteenth century. Since computing eigenvalues and vectors is essentially more complicated than solving linear systems, it is not surprising that highly significant developments in this area started with the introduction of electronic computers around 1950. In the early decades of this century, however, important theoretical developments had been made from which computational techniques could grow. Research in this area of numerical linear algebra is very active, since there is a heavy demand for solving complicated problems associated with stability and perturbation analysis for practical applications.