A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, A(λ)x = 0, is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear function A(λ) and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear function A(λ) has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independently of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small-and large-scale numerical examples are included. From the numerical experiments we can recommend two variants of the algorithm for solving the nonlinear eigenvalue problem.

Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.

Abstract. Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular, the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors, for which a MATLAB code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that are pointwise convergent as the degrees of the numerator and denominator increase to ∞, unlike traditional Padé approximants, which converge only in measure or capacity.

Abstract. A novel parallel algorithm for the integration of linear initial-value problems is proposed. This algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems. An overlapping time-domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems, and a near-optimal Krylov method is used for the fast exponential integration of the homogeneous subproblems. We present an error analysis and discuss the parallel scaling of our algorithm. The efficiency of this approach is demonstrated with numerical examples.

Abstract. When using the Arnoldi method for approximating f (A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments.

Abstract. Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided.Key words. rational Krylov decomposition, inverse eigenvalue problem, rational approximation AMS subject classifications. 15A22, 65F15, 65F18, 30E10 DOI. 10.1137/140998081 1. Introduction. Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. Rational Krylov spaces were initially proposed by Ruhe in the 1980s for the purpose of solving large sparse eigenvalue problems [37,39,40]. Since then many more applications have been found in model order reduction [22,17], large-scale matrix functions and matrix equations [13,15,1,26,27], and nonlinear eigenvalue problems [41,30,47,28], to name a few.In this paper we study various algebraic properties of rational Krylov spaces, using as a starting point a generalized rational Krylov decomposition

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