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The power method

The power method (see [5]) consists of the iterative scheme defined below

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

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

where the superscript (k) specifies the iteration number, the subscript i indicates the ith element of a vector and we choose the initial eigenvector $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ such that $\Vert\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}\Vert _{\infty} = 1$.

The mathematical properties of the eigenvalue problem given by theorems 1.5 and 1.2 ensure that (provided that $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ has a non-zero component in the direction of $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$)

$$\begin{eqnarray*} \lim\limits_{k \rightarrow \infty} L^{(k)} & = & \lambda_0 \\... ...x}^{(k)} & = & \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex} \end{eqnarray*}$$

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

where $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ is the eigenvector belonging to eigenvalue $\lambda_0$. In other words we are sure that the power method converges to the fundamental flux mode, ie converges to the eigensolution corresponding to the largest eigenvalue of the system.

In practical calculations both in the case of solving the diffusion equation and in general, experience shows that by comparing the convergence of the eigenvector ( $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(k)}$) and the eigenvalue (L^(k)) it is the eigenvector which has the slowest convergence. It is therefore natural to apply a convergence criterion on the eigenvector, ie a criterion of the form

$$\Vert \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(k)}... ...line{\phi}{}\hspace{0.15ex}^{(k-1)} \Vert _\infty \le \epsilon$$ (2.2)

where $\epsilon$ is expressing a measure for the desired accuracy^2.2. The asymptotic convergence rate of the power method is dependent on the ratio $(\vert\lambda_1\vert/\lambda_0)$ in the way that

$$\frac{E_{k+1}}{E_{k}} = \frac{\vert \lambda_1 \vert}{\lambda_0}$$ (2.3)

where $\lambda_1$ is the eigenvalue with the next largest modulo, $\lambda_0$ is the largest eigenvalue and E_k is defined by

$$E_{k} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscr... ...hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex} \Vert _{\infty}$$ (2.4)

where $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ is the exact eigenvector. A small ratio $(\vert\lambda_1\vert/\lambda_0)$ assures a fast convergence but if the two largest eigenvalues lie close together we have a slow convergence. Also, the number of iterations needed for accomplishing a given accuracy $\epsilon$ depends on the linear dependence of the initial guess $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ and the true eigenvector $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$--if they are strongly dependent, ie if $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ closely resembles $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ we need fewer iterations. Since we know (see Theorem 1.2) that $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ consists of all positive elements an obvious choice of the initial flux vector $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)}$ is

$$\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(0)} = \underbrace{[ 1,1,\ldots,1]^T}_{(K-2)G \mbox{ elements}}$$ (2.5)

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