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Tests involving the inverse power method

To test the performance of the inverse power method we ran the power method with a low accuracy criterion, say, $\epsilon = 10^{-2}$ in order to obtain a reasonable accurate estimate for the largest eigenvalue. This estimate was then used as the $\kappa$-input for the inverse power method (see section 2.3) which was run with a very strict accuracy criterion on the order of $\epsilon = 10^{-14}$.

The success of the described procedure for finding a suitable value of $\kappa$ depends on the eigenvalue approximation obtained from the power method. According to Theorem 1.4 the used eigenvalue has to satisfy

$$\lambda^{(k)}_{\mbox{\protect\scriptsize power\_method}} > \lambda_0$$ (4.10)

ie the exact eigenvalue has to be approached from above in order to guarantee the success of the simple forward elimination-backward substitution (see p. ).

From an extensive number of tests both on the test reactor and on reactor assemblies used in real power reactors we found that the demand (4.12) is in fact satisfied in all these tests if we choose a reasonably strict convergence criterion for the power method iterations on the order of $\epsilon \le 10^{-2}$ (see for instance Figure 3.6).

In Figures 4.14 and 4.15 we show the results involving the super-critical reactor assembly. It is seen that as with the power method the eigenvector error is larger that the eigenvalue error. Furthermore, we observe a highly improved convergence rate of the inverse power method compared to the power method. This high convergence rate arises from the fact that we have (see section 2.3)

$$\frac{E_{k+1,{\mbox{\protect\scriptsize exact}}}}{E_{k,{\mb... ...\vert \lambda_1 - \kappa \vert}{\vert\lambda_0 - \kappa\vert}$$ (4.11)

where $\lambda_1$ is the eigenvalue with the next largest modulo, $\lambda_0$ is the largest eigenvalue and $\kappa$ is the estimate for $\lambda_0$. From Figure 4.15 we estimate

$$\vert\lambda_0 - \kappa\vert \approx 2.1 \cdot 10^{-3}$$ (4.12)

and using the exact eigenvalues for the super-critical reactor assembly (see Table 4.1 we obtain

$$\frac{\vert\lambda_0 - \kappa\vert}{\vert \lambda_1 - \kapp... ...c{ 2.1 \cdot 10^{-3}}{ \vert.9277 - 1.076\vert} \approx 0.014$$ (4.13)

From the last part of the iterations in Figure 4.14 (apart from the last two iteration points which are influenced by the limited machine accuracy) we obtain the following approximation to the ratio

$$\left( \frac{ 10^{-15}}{1.0} \right)^{1/8.069} \approx 0.014$$ (4.14)

Upon comparing the two values we can conclude that the implementation of the inverse power method is working properly.

Finally, for the sub-critical reactor assembly we calculated the data depicted in Figures 4.16 and 4.17. As we can see there is no noteworthy difference in the sub-critical results as compared to the super-critical reactor assembly.

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