arXiv subjects

Daan Huybrechs

Publications and source records attributed to Daan Huybrechs.

46 records · Page 3Linked to original sources

arXiv2016

Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^α e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, α> -1, q_m > 0$. The classical Laguerre polynomials correspond to $Q(x)=x$. The computation of higher-order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full description seems to be lacking in literature. However, this information is implicitly available in the work of Vanlessen, based on a non-linear steepest descent analysis of an associated so-called Riemann--Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higher-order terms in the asymptotic expansions of Laguerre and Laguerre-type polynomials. This effort is similar to the case of Jacobi and Jacobi-type polynomials in a previous paper. We supply an implementation with explicit expansions in four different regions of the complex plane. These expansions can also be extended to Hermite-type weights of the form $\exp(-\sum_{k=0}^m q_k x^{2k})$ on $(-\infty,\infty)$, and to general non-polynomial functions $Q(x)$ using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules in a lower computational complexity than based on the recurrence relation, and with improved accuracy for large degree. They are also of interest in random matrix theory.

math.NA↗

arXiv2015

Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function $w(x)=(1-x)^α(1+x)^βh(x)$, $α,β>-1$ and $h(x)$ a real, analytic and strictly positive function on $[-1,1]$. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann--Hilbert formulation and the Deift--Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in $\mathcal{O}(n)$ operations, rather than $\mathcal{O}(n^2)$ based on the recurrence relation.

cs.MS↗

arXiv2015

Fast Algorithms for the computation of Fourier Extensions of arbitrary length

Fourier series of smooth, non-periodic functions on $[-1,1]$ are known to exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of overcoming these problems is by using a Fourier series on a larger domain, say $[-T,T]$ with $T>1$, a technique called Fourier extension or Fourier continuation. When constructed as the discrete least squares minimizer in equidistant points, the Fourier extension has been shown shown to converge geometrically in the truncation parameter $N$. A fast ${\mathcal O}(N \log^2 N)$ algorithm has been described to compute Fourier extensions for the case where $T=2$, compared to ${\mathcal O}(N^3)$ for solving the dense discrete least squares problem. We present two ${\mathcal O}(N\log^2 N )$ algorithms for the computation of these approximations for the case of general $T$, made possible by exploiting the connection between Fourier extensions and Prolate Spheroidal Wave theory. The first algorithm is based on the explicit computation of so-called periodic discrete prolate spheroidal sequences, while the second algorithm is purely algebraic and only implicitly based on the theory.

math.NA↗

arXiv2015

Fast and highly accurate computation of Chebyshev expansion coefficients of analytic functions

Chebyshev expansion coefficients can be computed efficiently by using the FFT, and for smooth functions the resulting approximation is close to optimal, with computations that are numerically stable. Given sufficiently accurate function samples, the Chebyshev expansion coefficients can be computed to machine precision accuracy. However, the accuracy is only with respect to absolute error, and this implies that very small expansion coefficients typically have very large relative error. Upon differentiating a Chebyshev expansion, this relative error in the small coefficients is magnified and accuracy may be lost, especially after repeated differentiation. At first sight, this seems unavoidable. Yet, in this paper, we focus on an alternative computation of Chebyshev expansion coefficients using contour integrals in the complex plane. The main result is that the coefficients can be computed with machine precision relative error, rather than absolute error. This implies that even very small coefficients can be computed with full floating point accuracy, even when they are themselves much smaller than machine precision. As a result, no accuracy is lost after differentiating the expansion, and even the 100th derivative of an analytic function can be computed with near machine precision accuracy using standard floating point arithmetic. In some cases, the contour integrals can be evaluated using the FFT, making the approach both highly accurate and fast.

math.NA↗

arXiv2013

Zero distribution of complex orthogonal polynomials with respect to exponential weights

We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights exp(-nV(z)) along contours in the complex plane. We are especially interested in the question under which circumstances the zeros of the orthogonal polynomials accumulate on a single analytic arc (one cut case), and in which cases they do not. In a family of cubic polynomial potentials V(z) = - iz^3/3 + iKz, we determine the precise values of K for which we have the one cut case. We also prove the one cut case for a monomial quintic V(z) = - iz^5/5 on a contour that is symmetric in the imaginary axis.

math.CA↗

arXiv2013

On the numerical stability of Fourier extensions

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation. In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.

math.NA↗

arXiv2012

A Gaussian quadrature rule for oscillatory integrals on a bounded interval

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{iωx}$ on the interval $[-1,1]$. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency $ω$. However, accuracy is maintained for all values of $ω$ and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as $ω\to 0$. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are always well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.

math.NA↗

arXiv2012

Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcal O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an ${\mathcal O}(n)$ computational scheme for barycentric weights.

math.NA↗

arXiv2012

On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and exponentially fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyze and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and π. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation.

math.NA↗

arXiv2011

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{iωt})(x)=-int_{0}^{\infty}e^{iωt}\frac{f(t)}{t-x}dt,\qquad ω>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $ω$ are derived for each fixed $x\geq 0$, which clarify the large $ω$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $ω$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.

math.CA↗