Q-linear convergence (${\mathrm{limsup}}_k\Vert e_{k+1}\Vert /\Vert e_k\Vert <1$)

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.