Methods of numerical integration of oscillatory functions by the DE-formula with the Richardson extrapolation

摘要

In 1978, a method for numerical integration of oscillatory functions by means of the DE-formula and the Richardson extrapolation was introduced by Toda and Ono, and yet, its usefulness was proved only experimentally. In this paper, theoretical considerations are bestowed on the usefulness of this Toda and Ono's method. Particularly, a class of integration problems is given for which Toda and Ono's method is efficient. An alternative method is also proposed along the same line as Toda and Ono's method. The usefulness of the proposed method is considered theoretically. Finally, it is demonstrated through numerical experiments that the proposed method surpasses Toda and Ono's method in efficiency.