Developing a methodology for the non-destructive analysis of British soft-paste porcelain

Soft-paste porcelain was produced in Britain in great quantities between the mid-18th and early 19th centuries. Due to industrial secrecy and the complexities of creating a product that would survive high-temperature firing, a range of paste recipes was employed by dozens of factories. This has resulted in an array of porcelains which vary in their elemental composition and mineralogy. This research carries out a meta-analysis of the published data for porcelain bodies and glazes and concludes that some discrimination can be achieved using the major and minor elemental composition of the bodies, and that for the glazes intra-factory variation is often greater than inter-factory variation in composition. A pilot investigation of the trace elemental composition of British porcelain is carried out using Laser Ablation Inductively Coupled Plasma Mass Spectroscopy, which finds compositional groups corresponding to different sources of clay and silica raw materials. In the interests of preserving intact objects, there is recognised a need for a non-destructive method for analysing British porcelain, in order to provenance and date objects. Such a method would rely on data from the surface of the object, which is typically covered by glaze and over-glaze coloured enamels, and this research demonstrates that the formulae used for the glaze and enamels are in some cases characteristic of the factory, or workshop, and period at which they were created. Hand-Held XRF analysis is used to analyse the glaze, underglaze blue and polychrome enamels on a selection of porcelain objects from different factories, and compositional traits are identified that allow some factories and periods to be distinguished. Glass standards are developed, which are representative of the glaze and enamel composition, and which could allow X-ray fluorescence (XRF) data to be calibrated for fully quantitative results

Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Computation of parabolic cylinder functions having complex argument

Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main main methods can be complemented with Maclaurin series and a Poincar\'e asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with $5\times 10^{-13}$ relative accuracy in double precision floating point arithmetic

Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order $\mu$ and degree $-\frac12+i\tau$, where $\tau$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ and ${Q}^{\mu}_{-\frac12+i\tau}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1<x<1$ (see \cite{gil:2009:con}, \cite{gil:2012:cpc}).In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ for $x>1$, the function $\Re\left\{e^{-i\pi \mu} {{Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.Comment: To be published in Numerical Algoritm

Computation of the coefficients appearing in the uniform asymptotic expansions of integrals

The coefficients that appear in uniform asymptotic expansions for integrals are typically very complicated. In the existing literature the majority of the work only give the first two coefficients. In a limited number of papers where more coefficients are given the evaluation of the coefficients near the coalescence points is normally highly numerically unstable. In this paper, we illustrate how well-known Cauchy type integral representations can be used to compute the coefficients in a very stable and efficient manner. We discuss the cases: (i) two coalescing saddles, (ii) two saddles coalesce with two branch points, (iii) a saddle point near an endpoint of the interval of integration. As a special case of (ii) we give a new uniform asymptotic expansion for Jacobi polynomials $P_n^{(\alpha,\beta)}(z)$ in terms of Laguerre polynomials $L_n^{(\alpha)}(x)$ as $n\to\infty$ that holds uniformly for $z$ near $1$. Several numerical illustrations are included.Comment: 18 page

Computation of Asymptotic Expansions of Turning Point Problems via Cauchy's Integral Formula: Bessel Functions

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville´Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.The authors acknowledge support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). A.G. and J.S. acknowledge support from Ministerio de Economía y Competitividad, project MTM2012-34787. A.G. acknowledges the Fulbright/MEC Program for support during her stay at SDSU. J.S. acknowledges the Salvador de Madariaga Program for support during his stay at SDSU