What does it mean to *accelerate* the convergence of an iterative method?

Question

First time reader of some numerical analysis papers, and I haven't come across a straight forward definition of acceleration of convergence. I'm guessing it means to improve convergence by a certain standard of magnitude. But what is the criteria for improvement that qualifies as acceleration; is there a formal definition?

Answer 1

Let $\{x_n\}_{n=1~}^\infty$ be as sequence, such that $$x_n \rightarrow \alpha, \quad n \rightarrow \infty, \quad n \in \mathbb{N},$$ and let $\{\tilde{x}_n\}_{n=1}^\infty$ be a sequence obtained from $\{x_n\}_{n=1~}^\infty$. We say, that $\{\tilde{x}_n\}$ converges faster than $\{ x_n\}$ to $\alpha$ provided $$ \frac{\tilde{x}_n- \alpha}{x_n - \alpha} \rightarrow 0, \quad n \rightarrow \infty, \quad n \in \mathbb{N}.$$

These notes are not a bad place to start reading about the acceleration of sequences.