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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