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Folkmar Bornemann

Publications and source records attributed to Folkmar Bornemann.

27 records · Page 2Linked to original sources
arXiv2012

On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw-Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n^{-s-1}) for some s > 0, Clenshaw-Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.

math.NA↗
arXiv2010

On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review

In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) beta-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the k-th largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.

math.PR↗
arXiv2010

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.

math.NA↗
arXiv2009

Asymptotic Independence of the Extreme Eigenvalues of GUE

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n \times n Gaussian Unitary Ensemble in the large matrix limit n \to \infty. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n^{-2/3}/4σ^2, where σ^2 denotes the variance of the Tracy--Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random hermitian matrix ensembles in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian Unitary Ensemble statistically from other Wigner ensembles.

math.PR↗
arXiv2008

On the Numerical Evaluation of Fredholm Determinants

Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss-Legendre or Clenshaw-Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy1 processes for the first time.

math.NA↗
arXiv2005

A Model for Understanding Numerical Stability

We present a model of roundoff error analysis that combines simplicity with predictive power. Though not considering all sources of roundoff within an algorithm, the model is related to a recursive roundoff error analysis and therefore capable of correctly predicting stability or instability of an algorithm. By means of nontrivial examples, such as the componentwise backward stability analysis of Gaussian elimination with a single iterative refinement step, we demonstrate that the model even yields quantitative backward error bounds that show all the known problem-dependent terms (with the exception of dimension-dependent constants, which are the weak spot of any a priori analysis). The model can serve as a convenient tool for teaching or as a heuristic device to discover stability results before entering a further, detailed analysis.

math.NA↗
arXiv2004

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

math.NA↗
arXiv2002

Runge-Kutta methods, trees, and Mathematica

A simple and elementary proof of Butcher's theorem on the order conditions of Runge-Kutta methods is presented. It is based on a recursive definition of rooted trees and avoids combinatorial tools such as labelings and Faa di Bruno's formula. This strictly recursive approach can easily and elegantly be implemented using modern computer algrebra systems like Mathematica. The full but short source code is included and applied to some instructive examples.

math.NA↗