Bernhard Beckermann

Silviu-Ioan Filip, Yuji Nakatsukasa, Lloyd N. Trefethen et al.

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.

Bernhard Beckermann, Alex Townsend

Matrices with displacement structure such as Pick, Vandermonde, and Hankel matrices appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the $k$th singular value of a real $n\times n$ positive definite Hankel matrix, $H_n$, is bounded by $Cρ^{-k/\log n}\|H\|_2$ with explicitly given constants $C>0$ and $ρ>1$, where $\|H_n\|_2$ is the spectral norm. This means that a real $n\times n$ positive definite Hankel matrix can be approximated, up to an accuracy of $ε\|H_n\|_2$ with $0<ε<1$, by a rank $\mathcal{O}(\log n\log(1/ε) )$ matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, Löwner, and certain Krylov matrices.

Catalin Badea, Bernhard Beckermann

This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets, introduced by John von Neumann, offer a possibility to estimate the norm of functions of matrices in terms of the sup-norm of the function. Examples of such spectral sets include the numerical range or the pseudospectrum of a matrix, discussed in Chapters 16 and 18. Estimating the norm of functions of matrices is an essential task in numerous fields of pure and applied mathematics, such as (numerical) linear algebra, functional analysis, and numerical analysis. More specific examples include probability, semi-groups and existence results for operator-valued differential equations, the study of numerical schemes for the time discretization of evolution equations, or the convergence rate of GMRES (Section 41.7). The notion of spectral sets involves many deep connections between linear algebra, operator theory, approximation theory, and complex analysis.

Bernhard Beckermann, George Labahn, Ana C. Matos

In this paper we consider the problem of working with rational functions in a numeric environment. A particular problem when modeling with such functions is the existence of Froissart doublets, where a zero is close to a pole. We discuss three different parameters which allow one to monitor the absence of Froissart doublets for a given general rational function. These include the euclidean condition number of an underlying Sylvester-type matrix, a parameter for determing coprimeness of two numerical polynomials and bounds on the spherical derivative. We show that our parameters sharpen those found in a previous paper by two of the autours.

Hedi Joulak, Bernhard Beckermann

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several convergence theorems for these quadrature formulae.

Bernhard Beckermann, Clara Mertens, Raf Vandebril

Matrices whose adjoint is a low rank perturbation of a rational function of the matrix naturally arise when trying to extend the well known Faber-Manteuffel theorem, which provides necessary and sufficient conditions for the existence of a short Arnoldi recurrence. We show that an orthonormal Krylov basis for this class of matrices can be generated by a short recurrence relation based on GMRES residual vectors. These residual vectors are computed by means of an updating formula. Furthermore, the underlying Hessenberg matrix has an accompanying low rank structure, which we will investigate closely.

Bernhard Beckermann, Thomas Helart

In this paper we show that the weighted Bernstein-Walsh inequality in logarithmic potential theory is sharp up to some new universal constant, provided that the external field is given by a logarithmic potential. Our main tool for such results is a new technique of discretization of logarithmic potentials, where we take the same starting point as in earlier work of Totik and of Levin \& Lubinsky, but add an important new ingredient, namely some new mean value property for the cumulative distribution function of the underlying measure. As an application, we revisit the work of Beckermann \& Kuijlaars on the superlinear convergence of conjugate gradients. These authors have determined the asymptotic convergence factor for sequences of systems of linear equations with an asymptotic eigenvalue distribution. There was some numerical evidence to let conjecture that the integral mean of Green functions occurring in their work should also allow to give inequalities for the rate of convergence if one makes a suitable link between measures and the eigenvalues of a single matrix of coefficients. We prove this conjecture , at least for a class of measures which is of particular interest for applications.

Bernhard Beckermann, Daniel Kressner, Marcel Schweitzer

We consider the task of updating a matrix function $f(A)$ when the matrix $A\in{\mathbb C}^{n \times n}$ is subject to a low-rank modification. In other words, we aim at approximating $f(A+D)-f(A)$ for a matrix $D$ of rank $k \ll n$. The approach proposed in this paper attains efficiency by projecting onto tensorized Krylov subspaces produced by matrix-vector multiplications with $A$ and $A^*$. We prove the approximations obtained from $m$ steps of the proposed methods are exact if $f$ is a polynomial of degree at most $m$ and use this as a basis for proving a variety of convergence results, in particular for the matrix exponential and for Markov functions. We illustrate the performance of our method by considering various examples from network analysis, where our approach can be used to cheaply update centrality and communicability measures.