On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw-Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n^{-s-1}) for some s > 0, Clenshaw-Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.Comment: 7 pages, the figure of the revision has an unsymmetric example, to appear in SIAM J. Numer. Ana

On quadrature of Bessel transformations

AbstractA method for integral transformations of highly oscillatory functions, Bessel functions, is presented. It is based on the Filon-type method and the decay of the error can be increased as α increases. The effectiveness and accuracy of the quadrature is tested for both large arguments and higher orders of Bessel functions in the case where the orders are nonnegative integers

A convergence analysis of block accelerated over-relaxation iterative methods for weak block H-matrices to partition π

AbstractThe aim of this paper is to establish the convergence of the block iteration methods such as the block successively accelerated over-relaxation method (BAOR) and the symmetric block successively accelerated over-relaxation method (BSAOR): Let A∈Cπ,nm,m be a weak block H-matrix to partition π, then for 0⩽r⩽ω⩽21+ρ(|BJ(A)|),ρ(BLr,ω)⩽|1-ω|+ωρ(|BJ(A)|),ρ(BSr,ω)⩽[|1-ω|+ωρ(|BJ(A)|)]2,and exact convergence and divergence domains for the block SOR and block SSOR iterative methods are obtained as it has been obtained to H-matrices. Based on these results, the main results in Bai [Parallel Computing 25 (1999)] and Cvetković [Appl. Numer. Math. 41 (2002)] can be improved

Barycentric Interpolation Based on Equilibrium Potential

A novel barycentric interpolation algorithm with a specific exponential convergence rate is designed for analytic functions defined on the complex plane, with singularities located near the interpolation region, where the region is compact and can be disconnected or multiconnected. The core of the method is the efficient computation of the interpolation nodes and poles using discrete distributions that approximate the equilibrium logarithmic potential, achieved by solving a Symm's integral equation. It takes different strategies to distribute the poles for isolated singularities and branch points, respectively. In particular, if poles are not considered, it derives a polynomial interpolation with exponential convergence. Numerical experiments illustrate the superior performance of the proposed method