Convergence acceleration by a method of intercalation

Abstract

Many convergence acceleration techniques function successfully when applied to series of real terms which alternate in sign, but are unsuccessful when applied to series of terms having the same sign. Given a series Σuv of terms having the same sign and whose sum U is required, it is often possible to construct a series of the form u 0-v0+u1-v1+..., where Σvv is a series whose terms have the same sign as the {uv} and whose sum V is known, which is amenable to transformation. The required estimate of U is then recovered from the transformed estimate of U-V and the known value of V. Use of this artifice is illustrated with reference to the {combining triple dot above}-algorithm, two numerical examples being given. © 1972 Springer-Verlag.