Mathematics-Online lexicon: Von Mises Iteration

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	Mathematics-Online lexicon:
	Von Mises Iteration

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overview

The von Mises iteration applies powers of a matrix to a start vector

. The resulting normalized sequence

$\displaystyle u_n = A^n x / \left\Vert A^n x \right\Vert _2$

will generally approximate a dominant eigenvector. Sufficient for convergence is that

is diagonalizable and has an eigenvalue $\lambda$ with largest absolute value. Then, for any vector

with a nontrivial component

in the eigenspace of $\lambda$ ,

$\displaystyle \left\Vert e^{- \mathrm{i} n \varphi} u_n - \frac{u }{\left\Vert... ...2} \right\Vert= O \left( \left\vert \varrho / \lambda \right\vert^n \right)$

with $e^{\mathrm{i} \varphi}=\lambda / \left\vert \lambda \right\vert$ and $\varrho$ an eigenvalue of

with second largest absolute value.

(Authors: Höllig/Pfeil/Walter)

Annotation:

Proof: Von Mises Iteration

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automatically generated 4/24/2007