Extended quadrature rules for oscillatory integrands

Authors:
KyungJoong Kim;Ronald Cools;L. Gr. Ixaru
Affiliations:
School of Mathematical Science, Seoul National University, San 56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, South Korea;Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium;Institute of Physics and Nuclear Engineering, Department of Theoretical Physics, P.O. Box MG-6, Magurele, Bucharest, R-76900, Romania
Venue:
Applied Numerical Mathematics
Year:
2003

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Abstract

We consider the integral of a function y(x), I[y] = ∫-11y(x)dx and its approximation by a quadrature rule of the form QN[Y] = Σk=1N(wk(0)y(xk) + wk(1)y(1)(xk)+...+wk(p)y(p)(xk)), i.e., by a rule which uses the values of both y and its derivatives up to p-th order at the nodes of the quadrature rule. We focus only on the case when the nodes are assumed known and present the procedure to calculate the weights. Two cases are actually examined: (i) y(x) is a polynomial and (ii) y(x) is an ω dependent function of the form y(x) = f1 (x) sin(ωx) + f2(x) cos(ωx) with smoothly varying f1 and f2. For the latter case, the weights wkj (j = 0, 1 ....., p) are ω dependent. A series of specific properties for this case is established and a numerical illustration is given.