superlinear convergence

Convergence faster than a fixed fraction per step (linear) but not quite the squaring speed of quadratic convergence -- the secant method's own rate.

A sequence converges superlinearly if $\dfrac{|p_{n+1}-p|}{|p_n-p|}\to0$ as $n\to\infty$ , but not necessarily at order $2$ . The secant method converges with order $\alpha=\dfrac{1+\sqrt5}{2}\approx1.618$ (the golden ratio) -- strictly between linear ( $\alpha=1$ ) and Newton's method's quadratic ( $\alpha=2$ ).

Examples

Broyden's method for systems also converges superlinearly, at the same golden-ratio-flavored rate as the secant method it generalizes.