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Experiment 3

In the previous experiment there was no sign of the influence of subspace iteration. In this experiment however we will clearly see the effect of the subspace iteration. For this example again two clusters of eigenvalues were chosen $ [1:100]$ and $ [1000:1009]$, one expects a clear view of the convergence of the subspace iteration in this case. Because the Lanczos-Ritz values approximate the extreme eigenvalues, it will at least take $ 20$ steps before the $ 10$ dominant eigenvalues are approximated. After these steps one can expect to see the convergence of the subspace iteration. The first figure (left of Figure 6.3) shows for each step $ j=1,2,\ldots,n-1$ in the algorithm the norms of the blocks $ S(i:n,1:i-1)$ for $ i=n-j:n$, the lines correspond to one particular submatrix, i.e., the norm of this submatrix is shown after every step in the algorithm. The second figure (right of Figure 6.3) is constructed in an analogous way as in Experiment 6.4.2. In Figure 6.4 it can be seen in which step the Ritz values approximate the most extreme eigenvalues well enough, this is also the point from which the convergence behavior starts in Figure 6.3.

Figure 6.3: Equally spaced eigenvalues in two clusters $ 1:100$ and $ 1000:1009$
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Figure 6.4: Lanczos behavior of equally spaced eigenvalues in two clusters $ 1:100$ and $ 1000:1009$
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It is clearly seen in Figure 6.3 that the subspace iteration starts with a small delay (as soon as the Lanczos-Ritz values approximate the dominant eigenvalues well enough the convergence behavior starts).


next up previous contents index
Next: Experiment 4 Up: Numerical experiments Previous: Experiment 2   Contents   Index
Raf Vandebril 2004-05-03