Reply to the “Comment on Reliable operations on oscillatory functions” by Dr. Ixaru: [Comput. Phys. Commun. 134 (2001)] Original Research Article

Stimulated by the ‘Comment’ of L. Ixaru [Comput. Phys. Commun. 134 (2001) previous paper], we investigate the relationships between the exponentially fitted numerical differentiation and Simpson quadrature of oscillatory functions [L. Ixaru, Comput. Phys. Commun. 105 (1997) 1] (LI), and the subsequently reported corresponding formulae [Gh. Adam, S. Adam, Comput. Phys. Commun. 125 (2000) 127] (A&A). We find that the A&A formulae achieve an optimum in the absolute sense, while the LI formulae satisfy a Rayleigh–Ritz optimum principle. As it concerns their practical interest, the A&A formulae are to be preferred whenever the problems to be solved involve harmonic series (trigonometric polynomials). However, if the oscillatory function is nothing but a black box for the user, then only the exponentially fitted formulae are available.