Difference between revisions of "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 | + | {{TEX|done}} |
+ | 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 | + | 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==== | ====References==== |
Latest revision as of 12:05, 13 August 2014
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
[1] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |
[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
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).
Rate of convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rate_of_convergence&oldid=13011