Is Gauss quadrature better than Clenshaw-Curtis?

We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of $\log((z+1)/(z-1))$ in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at $z=\infty$. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $[-1,1]$

Drugs research: an overview of evidence and questions for policy

In 2001 the Joseph Rowntree Foundation embarked upon a programme of research that explored the problem of illicit drugs in the UK. The research addressed many questions that were often too sensitive for the government to tackle. In many cases, these studies represented the first research on these issues. This study gives an overview of the projects in the programme. The topics covered include: * The policing of drug possession. * The domestic cultivation, purchasing and heavy use of cannabis. * Non-problematic heroin use, heroin prescription and Drug Consumption Rooms. * The impact of drugs on the family. * Drug testing in schools and in the workplac

Reviving the Method of Particular Solutions

Fox, Henrici and Moler made famous a "Method of Particular Solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant

Gaussian elimination as an iterative algorithm

Gaussian elimination (GE) for solving an $n \times n$ linear system of equations $Ax=b$ is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too

Large-scale computation of pseudospectra using ARPACK and eigs

ARPACK and its MATLAB counterpart, eigs, are software packages that calculate some eigenvalues of a large non-symmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000

X-Ray Absorption from the Milky Way Halo and the Local Group

Million degree gas is present at near-zero redshift and is due either to a gaseous Galactic Halo or a more diffuse but very massive Local Group medium. We can discriminate between these models because the column densities should depend on location in the sky, either relative to the Galaxy bulge or to the M31-Milky Way axis. To search for these signatures, we measured the OVII Kalpha absorption line strength toward 25 bright AGNs, plus LMC X-3, using XMM-Newton RGS archival data. The data are in conflict with a purely Local Group model, but support the Galactic Halo model. The strongest correlation is between the OVII equivalent widths and the ROSAT background emission measurement in the R45 band (0.4-1 keV), for which OVII emission makes the largest single contribution. This suggests that much of the OVII emission and absorption are cospatial, from which the radius of a uniform halo appears to lie the range 15-110 kpc. The present data do not constrain the type of halo gas model and an equally good fit is obtained in a model where the gas density decreases as a power-law, such as r^(-3/2). For a uniform halo with a radius of 20 kpc, the electron density would be 9E-4 cm^(-3), and the gas mass is 4E8 Msolar. The redshift of the four highest S/N OVII measurements is consistent with a Milky Way origin rather than a Local Group origin.Comment: 32 pages (14 figures); ApJ, in pres

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

Ten Digit Algorithms

This paper was presented as the A R Mitchell Lecture at the 2005 Dundee Biennial Conference on Numerical Analysis, 27 June 2005