I-acceleration convergence of double sequences

A sequence transformation T is a function T : (xk) ? (x? k) which maps a slowly convergent sequence to another sequence with better numerical properties. If limk xk = x and limk x? k = x? with rk and r? k as the truncated errors. Then we have xk = x + rk, x? k = x? + r? k . We say that the sequence (xk) converge more rapidly than the sequence (x? k) if lim k x? k ? x? xk ? x = lim k r? k rk = 0.