Rate of convergence

A characteristic of an iterative method that enables one to make a judgement on the dependence of the error of the method at the $n$-th iteration on the number $n$ (see [1]–[3]). For example, if $\|z^n\|\leq q^n\|z^0\|$, where $\|z^n\|$ is the norm of the error at the $n$-th iteration, while $q<1$, then one says that the method converges with the rate of a geometric progression with denominator $q$, while the value $-\ln q$ is called the asymptotic rate of convergence.

Given inequalities of the type $\|z^{n+1}\|\leq C\|z^n\|^k$, one speaks of a polynomial rate of convergence of order $k$ (for example, the quadratic rate of convergence of the Newton–Kantorovich iteration method, cf. Kantorovich process).

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Of course, one can speak of the rate of convergence of any process (not just iterative) in which convergence plays a role. See, in particular, Approximation of functions (and related articles).