The RKFIT Algorithm for Nonlinear Rational Approximation

Abstract

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 )}

Computing the Weighted Geometric Mean of Two Large-Scale Matrices and Its Inverse Times a Vector

“…Nevertheless, in some case the Gauss-Jacobi quadrature applied to (8) converges faster than the same method applied to (7). We analyse the convergence just for z ∈ R + , because we want to apply the formulae to matrices having positive real eigenvalues and the convergence of the quadrature formulae for matrices follows from the convergence of the same formulae for their eigenvalues.…”

“…For the rational Krylov methods, the poles are chosen according to either the adaptive strategy by Güttel and Knizhnerman [24] or the function rkfit from the rktoolbox [5], based on an algorithm by Berljafa and Güttel [6,7]. In our implementation, when the matrices are larger than 1000 × 1000, we get the poles by running rkfit on a surrogate problem of size 1000 × 1000 whose setup requires a rough estimate of the extrema of the spectrum of A −1 B.…”

Sampling-free parametric model reduction of systems with structured parameter variation

“…Substituting (22) into (21), which is then substituted into (20), yields The upper bound follows by taking norms on both sides.…”

Parallelization of the Rational Arnoldi Algorithm

“…This is a nonnormal matrix, and its eigenvalues are shown in Figure 2(a), together with the m = 16 poles used in this example. The poles are obtained using the RKFIT algorithm [5,6] and optimized for approximating exp(A)b, where the starting vector b has all its entries equal to 1. (A similar example is considered in [5, 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…”