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

Abstract. The RKFIT algorithm outlined in [M. Berljafa and S. Güttel, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 894-916] is a Krylov-based approach for solving nonlinear rational least squares problems. This paper puts RKFIT into a general framework, allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator. Furthermore, we derive a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and for root-finding. We also discuss similarities and differences between RKFIT and the popular vector fitting algorithm. A MATLAB implementation of RKFIT is provided, and numerical experiments, including the fitting of a multiple-input/multiple-output (MIMO) dynamical system and an optimization problem related to exponential integration, demonstrate its applicability.Key words. nonlinear rational approximation, least squares, rational Krylov method AMS subject classifications. 15A22, 65F15, 65F18, 30E10 DOI. 10.1137/15M10254261. Introduction. Rational approximation problems arise in many areas of engineering and scientific computing. A prominent example is that of system identification and model order reduction, where calculated or measured frequency responses of dynamical systems are approximated by (low-order) rational functions [20,26, 2,23,19]. Some other areas where rational approximants play an important role are analog filter design [7], time stepping methods [43], transparent boundary conditions [28,17], and iterative methods in numerical linear algebra (see, e.g., [32,42,18,27,33]). Here we focus on discrete rational approximation in the least squares (LS) sense.In its simplest form the weighted rational LS problem is the following: given data pairs {(λ i , f i )}

SUMMARYIn many scientific applications, the solution of nonlinear differential equations are obtained through the setup and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences, there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems, the current strategy amounts to solving each eigenproblem in isolation. We propose an alternative approach that exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (Chebyshev filtered subspace iteration (ChFSI)). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.

Abstract. Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.Key words. rational Krylov, orthogonalization, parallelization AMS subject classifications. 68W10, 65F25, 65G50, 65F15 DOI. 10.1137/16M10791781. Introduction. Rational Krylov methods have become an indispensable tool of scientific computing. Invented by Ruhe for the solution of large sparse eigenvalue problems (see, e.g., [26,27]) and frequently advocated for this purpose (see, e.g., [23,32] and [2, sect. 8.5]), these methods have seen an increasing number of other applications over the last two decades or so. Examples of rational Krylov applications can be found in model order reduction [11,16,17,22], matrix function approximation [10,13,19], matrix equations [3,9,24], nonlinear eigenvalue problems [20,21,31], and nonlinear rational least squares fitting [5,6].At the core of most rational Krylov applications is the rational Arnoldi algorithm, which is a Gram-Schmidt procedure for generating an orthonormal basis of a rational Krylov space. Given a matrix A ∈ C N,N , a vector b ∈ C N , and a polynomial q m of degree at most m and such that q m (A) is nonsingular, the rational Krylov space of order m is defined as