convergence rate

sequence converges rate

if

$$\underset{k\to \infty }{\lim }\frac{||e_{k+1}||}{||e_k|{|}^r}=C$$

where $C$ is finite nonzero constant, we can say it convergent with rate $r$
and

• $r=1$: linear (C < 1)
• $r>1$: super linear
• $r=2$: quadratic

also

• $|g^{\prime }(x^{\ast })|<1$ it can converge given acceptable interval
• $|g^{\prime }(x^{\ast })|>1$ it will diverge
• $|g^{\prime }(x^{\ast })|=0$ it’s at least quadratic