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Mathematics-Online lexicon:

Von Mises Iteration


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The von Mises iteration applies powers of a matrix to a start vector $ x$. 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 $ A$ is diagonalizable and has an eigenvalue $ \lambda$ with largest absolute value. Then, for any vector $ x$ with a nontrivial component $ u$ 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 $ A$ with second largest absolute value.
(Authors: Höllig/Pfeil/Walter)

see also:


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  automatically generated 4/24/2007