Optimal Gegenbauer quadrature over arbitrary integration nodes

This paper treats definite integrations numerically using Gegenbauer quadratures. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the developed numerical scheme employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the L∞L∞-norm and L2L2-norm, respectively. The key idea is to construct the Gegenbauer quadrature through discretizations at some optimal sets of points of the Gegenbauer–Gauss (GG) type in a certain optimality sense. We show that the Gegenbauer polynomial expansions can produce higher-order approximations to the definite integrals ∫−1xif(x)dx of a smooth function f(x)∈C∞[−1,1]f(x)∈C∞[−1,1] for the small range by minimizing the quadrature error at each integration point xixi through a pointwise approach. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Exact integrations are obtained for polynomials of any arbitrary degree nn if the number of columns in the developed Gegenbauer integration matrix (GIM) is greater than or equal to nn. The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates, faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments, including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods.