Numerical operations on oscillatory functions

We consider some typical numerical operations on functions (differentiation , integration, solving differential equations, interpolation) and show how the standard algorithms can be modified to become efficient when the functi ons are oscillatory, of the form y(x) = f(1)(x) sin(omegax) + f(2)(x) cos(o megax) where f(1)(x) and f(2)(x) are smooth functions. The expressions of t he parameters of the new formulae are written in a way which makes them tun ed also for functions of form y(x) = f(1)(x) sinh(lambdax) + f(2)(x) cosh(l ambdax). Our formulae only require the values of y at some points and those of omega or lambda and they tend to the classical formulae when omega or l ambda tends to zero. For the derivation we follow the exponential fitting t echnique introduced in a previous paper (L. Gr. Ixaru, Comput. Phys. Commun . 105 (1997), 1-19). We list the tuned expressions for the first and the se cond derivative, for the Simpson quadrature formula and for the Numerov alg orithm to solve differential equations. We also show how the Gauss quadratu re rule can be adapted and finally give a few tuned formulae for the interp olation. Numerical illustrations are presented for each case. Some open pro blems are also mentioned. (C) 2001 Elsevier Science Ltd. All rights reserve d.