Chapter 1 Mathematical Preliminaries

1.3 Algorithms and ConvergenceDefinition 1.8 Suppose {β n } ∞ n=1 is a sequence to converge to zero, and {α n } ∞ n=1 convergesto α. If exists a positive constant K such that|α n − α| ≤ K|β n |, for large n,then we say that {α n } ∞ n=1 converges to α with rate of convergence O(β n ). Writingα n = α + O(β n ). Usually, we use β n = 1 for some number p. We are generallynp interested in the largest value of p with α n = α + O(1/n p ).Definition 1.9 Suppose lim G(h) = 0 and lim F(h) = L. If exists a positive constanth→0 h→0K such that|F(h) − L| ≤ K|G(h)|, for sufficiently small h,then we write F(h) = L + O(G(h)). Usually, we use G(h) = h p for some number p. Weare generally interested in the largest value of p with F(h) = L + O(h p ).6