Optimizing Talbot's Contours for the Inversion of the Laplace Transform

Talbot's method for the numerical inversion of the Laplace Transform consists of numerically integrating the Bromwich integral on a special contour by means of the trapezoidal or midpoint rules. In this paper we address the issue of how to choose the parameters that define the contour, for the particular situation when parabolic PDEs are solved. In the process the well known subgeometric convergence rate O(e -c \sqrt N) of this method is improved to the geometric rate O(e -cN) with N the number of nodes in the integration rule. The value of the maximum decay rate c is explicitly determined. Numerical results involving two versions of the heat equation are presented. With the choice of parameters derived here, the rule-of-thumb is that to achieve an accuracy of 10 -l at any given time t, the associated elliptic problem has to be solved no more that l times.\ud \ud Supported by the National Research Foundation in South Africa under grant NRF528

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Parabolic and Hyperbolic Contours for Computing the Bromwich Integral

Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.\ud \ud JACW was supported by the National Research Foundation in South Africa under grant FA200503230001

A Numerical Methodology for the Painlevé Equations

The six Painlevé transcendents PI – PVI have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as ‘numerical mine fields’. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the PI equation. In later studies, we will concentrate on mathematical aspects of both the PI and the higher Painlevé transcendents

The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature

The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like $O(\varrho^{-2N})$, where $\varrho$ is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate $O(\varrho^{-N})$, i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.\ud \ud This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF

Degrees of adequacy: the disclosure of levels of validity in language assessment

The conceptualization of validity remains contested in educational assessment in general, and in language assessment in particular. Validation and validity are the subjective and objective sides of the process of building a systematic argument for the adequacy of tests. Currently, validation is conceptualized as being dependent on the validity of the interpretation of the results of the instrument. Yet when a test yields a score, that is a first indication of its adequacy or validity. As the history of validity theory shows, adequacy is further disclosed with reference to the theoretical defensibility (“construct validity”) of a language test. That analogical analytical disclosure of validity is taken further in the lingually analogical question of whether the test scores are interpretable, and meaningful. This paper will illustrate these various degrees of adequacy with reference mainly to empirical analyses of a number of tests of academic literacy, from pre-school level tests of emergent literacy, to measurements of postgraduate students’ ability to cope with the language demands of their study. Further disclosures of language test design will be dealt with more comprehensively in a follow-up paper. Both papers present an analysis of how such disclosures relate to a theoretical framework for responsible test design. Grade van toereikendheid: die ontsluiting van vlakke van geldigheid in taaltoetsingOpsommingOm geldigheid te konsepsualiseer bly 'n betwiste saak in opvoedkundige meting in die algemeen, en in taalassessering in die besonder. Geldigmaking en geldigheid kan respektiewelik opgevat word as die subjektiewe en objektiewe kante van die sistematiese argument wat gevoer kan word vir die toereikendheid van toetse. Tans word geldigmaking gekonseptualiseer as afhanklik van die interpretasie van die resultate van die instrument. Tog is dit so dat wanneer 'n toets 'n punt oplewer, dit 'n eerste aanduiding is van sy geldigheid. Soos die geskiedenis van geldigheidsteorie ook aantoon, word daardie toereikendheid verder ontsluit met verwysing na die teoretiese regverdiging (konstrukgeldigheid) van 'n taaltoets. Daardie logies-analitiese ontsluiting van geldigheid word verder geneem in die analogies linguale vraag: Is die toetsresultate interpreteerbaar en betekenisvol? Hierdie artikel illustreer hierdie verskillende grade van geldigheid met verwysing na empiriese analises van toetse van akademiese geletterdheid, vanaf voorskoolse toetse van ontluikende geletterdheid tot by metings van nagraadse studente se vermoë om die eise van akademiese diskoers te hanteer. Verdere ontsluitings van taaltoetsontwerp word vollediger hanteer in 'n opvolgartikel. Beide artikels bied 'n analise van hoe sulke ontsluitings verband hou met 'n teoretiese raamwerk vir verantwoordelike toetsontwerp. https://doi.org/10.19108/KOERS.84.1.245

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing $e^z$ in the the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly $O(3^{-N})$, where $N$ is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, $O(9.28903^{-N})$, can be achieved by using a different approach: best supremum-norm rational approximants to $e^z$ for $z\in (-\infty,0]$, following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody--Meinardus--Varga approximants by the method of Carathèodory-Fejèr approximation