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Wielandt Iteration

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The Wielandt iteration is an improvement of the von Mises method which generates a sequence of simultaneous approximations to an eigenvalue $ \lambda$ and a corresponding normalized eigenvector $ u$ of a matrix $ A$, starting from a sufficiently good initial guess. An iteration step is defined by

$\displaystyle \lambda_\ell$ $\displaystyle = u^{\operatorname t}_\ell A u_\ell$    
$\displaystyle v_\ell$ $\displaystyle = \left( A-\lambda_\ell E \right)^{-1} u_\ell$    
$\displaystyle u_{\ell+1}$ $\displaystyle = \frac{v_\ell}{\left\Vert v_\ell \right\Vert _2}$    

where, in an implementation, $ v_\ell$ is computed as a solution of a linear system.

For a simple eigenvalue of a symmetric matrix, the iteration converges cubically:

$\displaystyle \Delta_{\ell+1} \leq c \Delta_\ell^3 $

with $ \Delta_\ell=\max \left\{ \left\vert \lambda_\ell - \lambda \right\vert,\, \left\Vert u_\ell - \sigma_\ell u \right\Vert _2 \right\}$ and the sign $ \sigma_\ell$ chosen so that $ u_\ell^{\operatorname t} \left( \sigma_\ell u \right) \geq 0$.
(Authors: Höllig/Pfeil/Walter)

Example:

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  automatically generated 4/24/2007