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).

#### References[edit]

[1] |  N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)

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[2] |  G.I. Marchuk, "Methods of numerical mathematics" , Springer (1982) (Translated from Russian)
[3] |  A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)
[4] |  L.A. Hageman, D.M. Young, "Applied iterative methods" , Acad. Press (1981)
[5] |  J.F. Traub, "Iterative methods for the solution of equations" , Prentice-Hall (1964)

#### Comments[edit]

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).