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 $[-1,1]$

Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Hartree-Fock+BCS approach to unstable nuclei with the Skyrme force

We reanalyze the results of our extensive Hartree-Fock + BCS calculation from new points of view paying attention to the properties of unstable nuclei. The calculation has been done with the Skyrme SIII force for the ground and shape isomeric states of 1029 even-even nuclei ranging 2 <= Z <= 114. We also discuss the advantages of the employed three-dimensional Cartesian-mesh representation, especially on its remarkably high precision with apparently coarse meshes when applied to atomic nuclei. In Appendices we give the coefficients of finite-point numerical differentiation and integration formulae suitable for Cartesian mesh representation and elucidate the features of each formula and the differences from a method based on the Fourier transformation.Comment: 31 pages including 21 figures, Latex2

Developing and Analyzing Newton – C’otes Quadrature Formulae for Approximating Definite Integrals- AC++ Approach

In this paper, different Newton – C’otes quadrature formulae for the approximation of definite integrals and their error analysis are derived. The order of convergences of the methods is also derived and of these Newton – C’otes quadrature formulae, the Simpson’s 1/3 rule have been shown to have high order of convergence. Since the functionality of these numerical integration methods is practical only if we can use computer programs and applications to produce approximate solutions with acceptable errors within short period, C++ programs for the selected methods are written. These programs are used on the comparison of the Newton – C’otes quadrature formulae and the result obtained based on the inputs and outputs of the programs for different integrands. The results of these programs show that the convergence of the methods highly depends on the number of iterations. The results of different numerical examples show that for high accuracy of the trapezoidal rule computational effort is higher and round off errors with large number of iterations limit the accuracy. The results show that the Simpson’s 1/3 rule produces much more accurate solution than other methods even within small number of iterations. This shows that the error for Simpson’s rule 1/3 converges to zero faster than the error for the trapezoidal rule as the step size decreases. It is finally observed that Simpson’s 1/3 rule is much faster than the Trapezoidal and the Simpson’s 3/8 rules according to the results of the C++ programs

Sequential quadrature methods for RDO

Abstract This paper presents a comparative study between a large number of different existing sequential quadrature schemes suitable for Robust Design Optimization (RDO), with the inclusion of two partly original approaches. Efficiency of the different integration strategies is evaluated in terms of accuracy and computational effort: main goal of this paper is the identification of an integration strategy able to provide the integral value with a prescribed accuracy using a limited number of function samples. Identification of the different qualities of the various integration schemes is obtained utilizing both algebraic and practical test cases. Differences in the computational effort needed by the different schemes is evidenced, and the implications on their application to practical RDO problems is highlighted