View source for Rate of convergence

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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 [[#References|[1]]]–[[#References|[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|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|Kantorovich process]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,   "Methods of numerical mathematics" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Samarskii,   E.S. Nikolaev,   "Numerical methods for grid equations" , '''1–2''' , Birkhäuser  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Hageman,   D.M. Young,   "Applied iterative methods" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.F. Traub,   "Iterative methods for the solution of equations" , Prentice-Hall  (1964)</TD></TR></table>



====Comments====
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|Approximation of functions]] (and related articles).