arXiv subjects

Lloyd N. Trefethen

Publications and source records attributed to Lloyd N. Trefethen.

43 records · Page 3Linked to original sources

arXiv2017

The AAA algorithm for rational approximation

We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of Matlab and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name AAA stands for "adaptive Antoulas--Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a Matlab code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation.

math.NA↗

arXiv2016

Trigonometric Interpolation and Quadrature in Perturbed Points

The trigonometric interpolants to a periodic function $f$ in equispaced points converge if $f$ is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if $f$ is continuous. What if the points are perturbed? With equispaced grid spacing $h$, let each point be perturbed by an arbitrary amount $\le αh$, where $α\in [\kern .5pt 0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be be trouble for $α\ge 1/4$. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all $α<1/2$ if $f$ is twice continuously differentiable, with the convergence rate depending on the smoothness of $f$. More precisely it is enough for $f$ to have $4α$ derivatives in a certain sense, and we conjecture that $2α$ derivatives is enough. Connections with the Fejér--Kalmár theorem are discussed.

math.NA↗

arXiv2016

Multivariate polynomial approximation in the hypercube

A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial, but by the {\it Euclidean degree,} defined in terms of the 2-norm rather than the 1-norm of the exponent vector $\bf k$ of a monomial $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.

math.NA↗

arXiv2015

Chopping a Chebyshev Series

Chebfun and related software projects for numerical computing with functions are based on the idea that at each step of a computation, a function $f(x)$ defined on an interval $[a,b]$ is "rounded" to a prescribed precision by constructing a Chebyshev series and chopping it at an appropriate point. Designing a chopping algorithm with the right properties proves to be a surprisingly complex and interesting problem. We describe the chopping algorithm introduced in Chebfun Version 5.3 in 2015 after many years of discussion and the considerations that led to this design.

math.NA↗

arXiv2015

Extension of Chebfun to periodic functions

Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted.

math.NA↗

arXiv2000

Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-Hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a "bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices ("stochastic Toeplitz operators"), finite periodic matrices, and doubly infinite bidiagonal matrices ("stochastic Laurent operators").

cond-mat.stat-mech↗

arXiv2000

Threshold amplitudes for transition to turbulence in a pipe

Although flow in a circular pipe is stable to infinitesimal perturbations, it can be excited to turbulence by finite perturbations whose minimal amplitude shrinks as R goes to infinity (R = Reynolds number). Laboratory experiments have appeared to disagree with one another and with theoretical predictions about the dependence of this minimal amplitude on $R$, with published results ranging approximately from $R^{-1/4}$ to $R^{-3/2}$. Here it is shown that these discrepancies can be explained by the use of different definitions of amplitude by different authors. An attempt is made to convert the existing results to a uniform definition of amplitude, the nondimensionalized $L^2$ definition common in the theoretical literature. Although subtleties in the physics raise some questions, agreement appears to be reached on a minimal amplitude that scales as $R^{-3/2 \pm 0.3}$.

physics.flu-dyn↗