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114 CHAPTER 7. NUMERICAL EXPERIMENTS
norm of the difference between the computed smallest real eigenvalue and the actual
global minimum (computed by the Eig method on the explicit matrix A pλ ).
Table 7.6: Results using JD and JDQZ with the least-increments method: time
(mean±stdev), # MVs (mean±stdev), and the error
JD
JDQZ
# Time (s) #MVs Error Time (s) #MVs Error
1 0.541 ± 0.152 109.30 ± 6.789 5.63 × 10 −7 7.75 ± 6.83 2041.00 ± 1840 2.23 × 10 −1
2 0.183 ± 0.00204 36 2.09 × 10 −7 0.141 ± 0.00200 25 3.26 × 10 −6
3 0.894 ± 0.0715 91.60 ± 6.957 5.61 × 10 −8 1.41 ± 0.114 154.90 ± 13.35 1.40 × 10 −3
4 0.552 ± 0.0904 60.07 ± 9.721 6.99 × 10 −7 0.388 ± 0.0225 42.33 ± 1.759 2.79 × 10 −1
5 0.593 ± 0.0497 68.87 ± 5.680 9.12 × 10 −8 0.448 ± 0.0102 47.13 ± 0.743 1.62 × 10 −4
6 0.458 ± 0.00192 41 1.20 × 10 −14 0.458 ± 0.00214 41 1.30 × 10 −14
7 0.923 ± 0.00216 61 3.40 × 10 −11 0.912 ± 0.00294 61 4.27 × 10 −11
8 0.893 ± 0.00271 61 1.10 × 10 −13 0.883 ± 0.00152 61 1.84 × 10 −13
9 1.16 ± 0.00480 63 2.40 × 10 −14 1.16 ± 0.00359 63 4.80 × 10 −14
10 1.06 ± 0.00396 63 1.23 × 10 −13 1.06 ± 0.00321 63 1.94 × 10 −13
11 2.71 ± 0.00542 85 8.90 × 10 −12 2.68 ± 0.0115 85 4.61 × 10 −12
12 2.43 ± 0.00543 85 1.47 × 10 −11 2.40 ± 0.00376 85 8.22 × 10 −12
13 5.94 ± 0.0146 113 5.13 × 10 −9 5.98 ± 0.145 113 3.38 × 10 −9
14 5.43 ± 0.0354 113 9.13 × 10 −12 5.33 ± 0.0234 113 7.34 × 10 −12
15 16.0 ± 0.117 172 1.15 × 10 −11 15.6 ± 0.0542 172 7.58 × 10 −12
16 19.5 ± 0.0940 172 5.33 × 10 −12 19.2 ± 0.149 172 1.63 × 10 −12
17 62.9 ± 4.74 269.40 ± 6.957 3.57 × 10 −7 59.3 ± 1.46 266.10 ± 4.131 2.19 × 10 −7
18 69.6 ± 0.284 265 9.05 × 10 −8 68.6 ± 0.306 265 2.78 × 10 −8
19 148 ± 1.91 365 2.04 × 10 −8 144 ± 1.20 365 3.58 × 10 −9
20 174 ± 2.08 365 6.88 × 10 −10 170 ± 1.51 365 1.77 × 10 −10
21 1360 ± 12.1 666 1.53 × 10 −5 1350 ± 9.03 666 6.24 × 10 −6
22 1250 ± 7.94 666 2.46 × 10 −6 1230 ± 9.34 666 4.82 × 10 −7
To place the performance results obtained above in the appropriate context, all
cases are also tested with SOSTOOLS, the sum of squares toolbox for Matlab [90]. This
software returns a certified lower bound for a polynomial p. Each test case from
Table 7.4 is analyzed 20 times using SOSTOOLS with the default parameter settings.
The mean processing time as well as the mean error, the mean of the norm of the
difference between the computed smallest real eigenvalue of SOSTOOLS, and the actual
global minimum, are recorded. The results are given in Table 7.7.
From this table it is clear that for the majority of the test cases the SOSTOOLS software
approach is more efficient than the nD-system implementation presented here.
This is in accordance with the results described in [20]. In particular it appears that
in the SOSTOOLS approach the processing time increases less rapidly with increasing
complexity.
However, it should be noted that the error in the SOSTOOLS approach can be higher
than the error in the nD-systems implementation. In particular the test cases 17, 18,
21 and 22 can not accurately and reliably be solved by the SOSTOOLS software (at
least not when using the default parameter settings of SOSTOOLS). These large errors
can limit the actual application of this software on large problem instances.
Furthermore there are multivariate polynomials that are positive but can not be
written as a sum of squares [92]. The global minimum of these polynomials can not
be found directly using the SOSTOOLS approach. Such a limitation does not affect the
Stetter-Möller matrix method.