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7.3. PROJECTION TO STETTER STRUCTURE 115
Table 7.7: Results using SOSTOOLS: time (mean±stdev) and the error
# Time (s) Error
1 1.00 ± 1.58 3.35 × 10 −9
2 0.551 ± 0.00617 1.01 × 10 −9
3 0.675 ± 0.0150 8.35 × 10 −9
4 0.655 ± 0.00650 1.50 × 10 −8
5 0.597 ± 0.00662 6.29 × 10 −9
6 0.656 ± 0.00302 1.67 × 10 −8
7 1.16 ± 0.00980 1.51 × 10 −4
8 0.974 ± 0.0169 4.46 × 10 −8
9 0.691 ± 0.00570 8.24 × 10 −8
10 0.748 ± 0.00607 1.20 × 10 −7
11 1.56 ± 0.0254 5.05 × 10 −5
12 1.44 ± 0.0143 6.12 × 10 −6
13 3.11 ± 0.0185 3.22 × 10 −2
14 2.34 ± 0.0163 2.40 × 10 −6
15 1.58 ± 0.0153 3.86 × 10 −6
16 1.56 ± 0.0176 8.84 × 10 −7
17 14.2 ± 0.0206 1.65 × 10 +4 ± 3.77 × 10 −12
18 24.1 ± 0.0266 4.50 × 10 +5
19 4.48 ± 0.0112 8.58 × 10 −5
20 4.77 ± 0.00916 3.47 × 10 −5
21 19.4 ± 0.0105 1.77 × 10 −2
22 17.6 ± 0.00798 1.74 × 10 −3
7.3 Projection to Stetter structure
The effect of the projection of approximate eigenvectors to nearby approximate vectors
with Stetter structure on the convergence of the JD software is analyzed using
the first five polynomials of the set of polynomials mentioned in the previous section.
The result of this experiment is that with the settings presented in Table 7.5 and the
projection described in Section 6.4, the JD eigenvalue solver is no longer able to converge
to the desired eigenvalue, as can be seen in Figure 7.2 which shows the residual
r j before and after projection in the first test case for each iteration (see also [53]).
From this it is clear that the projection finds an improvement in each iteration.
However, this improvement is discarded by the eigenvalue solver in the next iteration.
A possible explanation for this may be the fact that the projected vector is no longer
contained in the original search space and therefore the improvement is eliminated
in the subsequent search space expansion and extraction of the new approximate
eigenpair. A possible solution for this problem is also given in Section 6.4. Nonetheless
these results strengthen the assumption that projection to Stetter structure can help
to speed up the convergence.
Further experimental analysis shows that if the expansion type of the JD eigenvalue
solver is changed from ‘jd’ to ‘rqi’ then the eigenvalue solver can converge to
the desired eigenvalue when projection is used. The Jacobi–Davidson method is an
extension of the RQI (Rayleigh Quotient Iteration) where in the first part of the convergence
process an inverse method is used to increase the convergence speed [58].
Table 7.8 shows the results of the first five test cases with the JD method with JD
expansion, the JD method with RQI expansion without projection and the JD method