A New Method for Approximating Improper Integrals
Henry L. Gray, Suojin Wang · SIAM Journal on Numerical Analysis · 1992
In this paper a new transformation referred to as the $G_n^{(m)} $-transform is introduced for the purpose of increasing the rate of convergence of a sequence to its limit. In particular, if the sequence is a sequence of partial integrals $F(x)$, then $G_n^{(m)} [F(x)]$ is shown to converge super-fast to $F(\infty )$ under very general conditions. The $G_n^{(m)} $-transform is shown to be closely related to Levin and Sidi’s D-transform [Appl. Math. Comp., 9 (1981), pp. 175–215] and the $B_n $-transform introduced by Gray, Atchinson, and McWilliams [SIAM J. Numer. Anal., 8 (1971), pp. 365–381]. Several examples are given.