Linear convergence ($|e_{k+1}|/|e_k|\to \rho \in (0,1)$)

An iteration of a map has linear convergence to its limit when its nonzero errors $e_k=x_k-\alpha$ satisfy $|e_{k+1}|/|e_k|\to \rho$ for some $0<\rho <1$. The error eventually shrinks by approximately the same factor at each step. This is exact order of convergence one, in contrast to quadratic convergence.

A sequence with errors $e_k=x_k-x^{\ast }$ converges Q-linearly in a specified norm if there is a constant $q<1$ with $\Vert e_{k+1}\Vert \le q\Vert e_k\Vert$ eventually. The limsup of consecutive nonzero error ratios is its asymptotic factor. This is stronger than merely having a bound $\Vert e_k\Vert \le C{q}^k$, which is called R-linear convergence. The norm and whether an iteration is one coordinate step or an entire sweep must be specified when quoting a factor.