order of convergence

How fast a sequence of approximations closes in on its limit -- whether each new error is a fixed fraction of the last, or shrinks much faster than that.

A sequence of approximations $p_n \to p$ converges with order $\alpha$ if there is a constant $\lambda$ with $\dfrac{|p_{n+1}-p|}{|p_n-p|^\alpha} \to \lambda$ as $n\to\infty$ ; $\alpha=1$ (with $\lambda<1$ ) is called linear convergence, $\alpha=2$ quadratic.

Examples

Bisection converges with order $\alpha=1$ ; Newton's method, near a simple root, converges with order $\alpha=2$ .