Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Abstract

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

Parallelization of the Rational Arnoldi Algorithm

“…Such a right-multiplication of the decomposition by R m is equivalent to changing the parameters (η j /ρ j , t j ) during the rational Arnoldi algorithm. The two RADs for Q m+1 (A, b, q m ) are essentially equal ; see [5,Def. 3.1] for a precise definition.…”

“…3.1] for a precise definition. In fact, the rational implicit Q theorem [5,Thm. 3.2] asserts that the RADs related to Q m+1 (A, b, q m ) are essentially uniquely determined by A, b, and the ordering of the poles of the decomposition.…”

“…For instance, in [27] the author refers to V j t j as the continuation vector, while in [23] the term is used to denote (ρ j A − η j I)V j t j . The terminology of "continuation combinations" is adopted in [5,27,31] for the vectors t j . With the notion of continuation pair we want to stress that there are two parts, a root and a vector, both of which are equally important.…”

“…Definition 2.1), the continuation pair (η m /ρ m , t m ) is optimal for (4) the rational implicit Q theorem [5,Thm. 3.2], which asserts that the new direction v m+1 we are interested in is predetermined by the parameters (A, b, q m ) defining Q m+1 (A, b, q m ).…”

“…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…”