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The fractional iteration method

The fractional iteration method (or inverse power method) is described by the following iteration scheme

$$\begin{eqnarray*} \mbox{Solve: } \vspace{1cm} (\hspace{0.2ex}\underline{\underl... ...1}{L^{(k)}} \hspace{0.2ex}\underline{z}{}\hspace{0.15ex}^{(k)} \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

where $\kappa$ is a approximation of the desired eigenvalue $\lambda_0$ and we choose the initial eigenvector $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ such that $\Vert\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}\Vert _{\infty} = 1$.

If we choose $\kappa$ such that it lies closest to $\lambda_0$, ie if

$$\min\limits_{i} \vert \lambda_i - \kappa \vert = \vert\lambda_0 - \kappa\vert$$ (2.6)

we have (provided that $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ has a non-zero component in the direction of $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$)

$$\begin{eqnarray*} \lim\limits_{k \rightarrow \infty} \hspace{0.2ex}\underline{\... ...\rightarrow \infty} L^{(k)} & = & \frac{1}{\lambda_0 - \kappa} \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

and the asymptotic convergence rate is dependent on the ratio

$$\frac{\vert \lambda_0 - \kappa \vert}{\min\limits_{i \ne 0}\vert \lambda_i - \kappa \vert}$$ (2.7)

ie the ratio between the largest and next largest eigenvalues in the eigenvalue spectrum shifted by $-\kappa$. More specifically the asymptotic convergence of the exact eigenvector error, E_k, defined by (2.6) is controlled by

$$\frac{E_{k+1}}{E_{k}} = \frac{\vert \lambda_0 - \kappa \vert}{\vert \lambda_1 - \kappa \vert}$$ (2.8)

where $\lambda_1$ is the eigenvalue with the next largest modulo, $\lambda_0$ is the largest eigenvalue and $\kappa$ is the approximation to $\lambda_0$. A small ratio assures a fast convergence and we therefore conclude that having a good estimate of $\lambda_0$ results in a fast converging process; this is verified by test calculations to be discussed later in this text.

We will use the convergence criterion (2.4) for the fractional iteration method as well.

It should be noted that it is possible to make an update of $\kappa$ in the iteration scheme (2.8) by utilizing that $L^{(k)} \rightarrow [\lambda_0 - \kappa]^{-1}$ in the following way

$$\kappa := \frac{1}{L^{(k)}} + \kappa$$ (2.9)

in order to get a faster convergence. The drawback is that we have to make a new factorization of $(\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}-\kappa \hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex})$--the common way of solving the equations--and this is a very costly operation in comparison with making additional iterations with the old factorization. Practical calculations show (see for instance section 4.3.2) that if one starts off with a good estimate for $\lambda_0$ it is not efficient to make an update of $\kappa$.

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