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

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

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

Automatic Detection of Outliers in Multibeam Echo Sounding Data

The data volumes produced by new generation multibeam systems are very large, especially for shallow water systems. Results from recent multibeam surveys indicate that the ratio of the field survey time, to the time used in interactive editing through graphical editing tools, is about 1:1. An important reason for the large amount of processing time is that users subjectively decide which soundings are outliers. There is an apparent need for an automated approach for detecting outliers that would reduce the extensive labor and obtain consistent results from the multibeam data cleaning process, independent of the individual that has processed the data. The proposed automated algorithm for cleaning multibeam soundings was tested using the SAX-99 (Destin FL) multibeam survey data [2]. Eight days of survey data (6.9 Gigabyte) were cleaned in 2.5 hours on an SGI platform. A comparison of the automatically cleaned data with the subjective, interactively cleaned data indicates that the proposed method is, if not better, at least equivalent to interactive editing as used on the SAX-99 multibeam data. Furthermore, the ratio of acquisition to processing time is considerably improved since the time required for cleaning the data was decreased from 192 hours to 2.5 hours (an improvement by a factor of 77)

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

Teaching/Learning: The Student Viewpoint

Learning is a vital aspect in the life of every individual. To some it comes, easily, but to others it does not. Why this happens depends on several interrelated factors. Among them are home environment and parental support, individual capabilities/potential including mental maturity and personal drive, and school/educational classroom methodology or procedures. The latter point is the focus of this paper

Resolving the Ripples (and a Mine): High-Resolution Multibeam Survey of Martha\u27s Vineyard ONR Mine Burial Program Field Area

In an effort to better understand the coastal processes responsible for the burial and exposure of small objects on the seafloor, the Office of Naval Research is sponsoring the Mine Burial Program. Among the field areas chosen for this program is the site of the Martha\u27s Vineyard Coastal Observatory (MVCO), a permanent instrumented node in 12 m of water about 500 m off the southern shore of Martha?s Vineyard. In support of the ONR program, several site surveys of the MVCO area have been conducted (see Goff et al); here we report the result of the most recent of these surveys, a very high-resolution multibeam survey aimed at establishing a detailed base map for the region and providing a baseline from which subsequent surveys can measure seafloor change In late July we conducted a five day survey of an approximately 3 x 5 km area surrounding the MVCO node using a Reson 8125 focused multibeam sonar aboard the SAIC survey vessel Ocean Explorer. The 8125 is a newly developed multibeam sonar that operates at 455 kHz and uses dynamic focusing to compensate for the curvature of the wavefront in the near-field. By using a relatively long array, the system can achieve very high spatial resolution (0.5 degree beam width) and with the dynamic focusing, can operate in the near field. The real constraint on resolution using this system is the ability to position the soundings and thus three kinematic DGPS base stations were established on Martha?s Vineyard and three kinematic receivers were used on the survey vessel. The kinematic GPS positioning is also critical to the ability to do repeat surveys with an accuracy high enough to resolve small (less than 10 cm) seafloor changes. Also to aid in our ability to accurately position repeat surveys, divers jetted sonar reflectors into the seafloor to act as fiducials. A super high-resolution (4 m overlap) survey was conducted in a small area surrounding the MVCO node and mine burial sites, a slightly lower resolution survey (12 to 25 m overlap) in a box approximately 1 x 1 km surrounding the ?target box? and a lower resolution survey (25 to 40 m line overlap) in a 3 x 5 km region surrounding the 1 x 1 km box. The Reson 8125 produced approximately 1 gigabyte of data per hour. The bathymetric resolution we were able to achieve was beyond our expectations. The node site and all diver-emplaced reflectors were clearly identified Most amazingly, we are able to resolve fields of individual ripples that are less than 2 cm height. Of particular relevance to the mine burial program was our ability to resolve an instrumented mine that had been deployed earlier by NRL. This mine is buried in a scour depression and is only a few centimeters proud above the base of the depression

Seafloor Characterization from Spatial Variation of Multibeam Backscatter vs. Grazing Angle

Backscatter vs. grazing angle, which can be extracted from multibeam backscatter data, depend on characteristics of the multibeam system and the angular responses of backscatter that are characteristic of different seafloor properties, such as sediment hardness and roughness. Changes in backscatter vs. grazing angle that are contributed by the multibeam system normally remain fixed over both space and time. Therefore, they can readily be determined and removed from backscatter data. The variation of backscatter vs. grazing angle due to the properties of sediments will vary from location to location, as sediment type changes. The sediment component of variability can be inferred using the redundant observations from different grazing angles in several small pieces of seafloor where the sediment property is uniform in any given piece of seafloor yet vary from one piece of the seafloor to another. Thanks to the multibeam survey (Roger Flood, State University of New York) at SAX 99 Project sponsored by Office of Naval Research (ONR), which had 800\% coverage in most of the survey area; there is a data set, which is suitable for investigating seafloor characterization. The investigation analyzed the spatial variation of the backscatter vs. grazing angle and compared that with ground truth sediment data. In this research, the 6.9 gigabytes raw multibeam data were cleaned using an automated outlier detection algorithm (Tianhang Hou, Lloyd Huff and Larry Mayer. 2001). Then, the surveyed area was equally divided into 52X78 rectangle working cells (4056), the side of each cell was about 20 meters. The backscatter vs. grazing angle of backscatter data for each cell is computed by averaging backscatter data by the corresponding beam numbers using all data with the same beam number from different survey lines. Systematic effects on the backscatter vs. grazing angle, caused by multibeam system hardware or software as well as system installation, were corrected in order to remove the asymmetric and skew effects. In order to easily evaluate the spatial variation of the backscatter vs. grazing angle, a graphic interface was developed. With a mouse click, the images based on different subsets of the data can be compared throughout the survey area. The subsets were created using specific beam numbers. These images for different beams show significant variations between nadir and off-nadir beams. These variations allow an interesting interpretation to be made of the images in light of seafloor characteristics, which were derived from ground truth data, such as sediment grain size, density and velocity

A six-coordinate aryl-germanium complex formed by the Kläui ligand

PhGeCl₃ reacts with Na{[OP(OEt)₂]₃CoCp} to give the six-coordinate complex PhCl₂Ge{[OP(OEt)₂]₃CoCp}, characterised spectroscopically and by an X-ray crystal structure determination which showed a firmly-attached tridentate ligand [Ge–O 1.973(2) Å]

X-ray Studies of Two Neutron Stars in 47 Tucanae: Toward Constraints on the Equation of State

We report spectral and variability analysis of two quiescent low mass X-ray binaries (X5 and X7, previously detected with the ROSAT HRI) in a Chandra ACIS-I observation of the globular cluster 47 Tuc. X5 demonstrates sharp eclipses with an 8.666+-0.01 hr period, as well as dips showing an increased N_H column. The thermal spectra of X5 and X7 are well-modeled by unmagnetized hydrogen atmospheres of hot neutron stars. No hard power law component is required. A possible edge or absorption feature is identified near 0.64 keV, perhaps an OV edge from a hot wind. Spectral fits imply that X7 is significantly more massive than the canonical 1.4 \Msun neutron star mass, with M>1.8 \Msun for a radius range of 9-14 km, while X5's spectrum is consistent with a neutron star of mass 1.4 \Msun for the same radius range. Alternatively, if much of the X-ray luminosity is due to continuing accretion onto the neutron star surface, the feature may be the 0.87 keV rest-frame absorption complex (O VIII & other metal lines) intrinsic to the neutron star atmosphere, and a mass of 1.4 \Msun for X7 may be allowed.Comment: 16 pages, 7 figures, accepted by Ap