An algorithm for Gauss-Romberg integration

WhenR ^(m) f is anm copy version of a quadrature ruleRf, the error functional satisfies an asymptotic expansion $$R^{(m)} f - If simeq d_2 h^2 + d_4 h^4 + ...,m = 1/h.$$ In the conventional form of Romberg Integration,Rf is the trapezoidal rule and early terms of this expansion are “eliminated.” For this purpose the Neville-Romberg algorithm is used to construct the conventionalT-table. IfRf is taken to be a ruleGf of polynomial degree 2t+1 the firstt terms in this expansion are in any case zero. A generalization of the Neville-Romberg algorithm is derived. This “eliminates” termsd _2s h ^2s ,s=t+1,t+2, ... An associatedG-table is defined and some of its properties are noted.