link to my thesis

7.4. PARALLEL COMPUTATIONS IN THE ND-SYSTEMS APPROACH 117
apart from the first test case, the results presented here can serve as an initial proof
of concept that projection to Stetter structure can be used to increase the convergence
speed of the eigenvalue solvers, at least if the initial estimate is accurate enough.
7.4 Parallel computations in the nD-systems approach
Although a parallel implementation of the least-increments method is not very promising
as explained in Section 5.3.4, we carried this out nevertheless. The results of computing
the global minimum of the test set of 22 polynomials using iterative solvers
and a parallel nD-system implementation are given in Table 7.9 (see also [53]). The
least-increments method is implemented here to compute each path in the nD-system
on a separate processor. The number of required paths are shown in the last column
of Table 7.4 as the number of terms of each polynomial. The number of processors
used here is 4. Table 7.9 shows clearly that the computing time is longer but the
required number of operator actions is roughly identical in this parallel environment
in comparison with the results in Table 7.6. However, it is also clear to see that
the increase in computation time for both the parallel JD and JDQZ method is not
that high when the number of paths and the size of the involved matrix/operator
increases: for example for the last 2 cases where the increase in computation time is
roughly 2 for both the iterative methods.
Table 7.9: Results for the parallel implementation of the nD-system for JD and JDQZ:
time (mean±stdev), # MVs (mean±stdev) and the error
JD
JDQZ
# Time (s) #MVs Error Time (s) #MVs Error
1 4.73 ± 0.289 120.3 ± 8.982 5.50 × 10 −7 14.8 ± 6.85 1991 ± 1314 5.04 × 10 −2
2 26 ± 0.131 36 5.52 × 10 −7 27.2 ± 0.557 41 1.90 × 10 −6
3 40 ± 0.522 88.67 ± 6.351 2.16 × 10 −7 37.8 ± 1.08 76.67 ± 6.429 3.79 × 10 −4
4 34.9 ± 1.38 59.33 ± 6.351 1.17 × 10 −7 31.6 ± 0.737 43 ± 3.464 4.65 × 10 −1
5 70.3 ± 1.39 77.67 ± 6.351 5.48 × 10 −8 64.8 ± 0.918 47.67 ± 0.577 8.61 × 10 −5
6 76.7 ± 3.08 52 ± 11 1.40 × 10 −14 73.6 ± 3.29 44.67 ± 6.351 1.40 × 10 −14
7 45.2 ± 0.824 61 7.29 × 10 −12 45.7 ± 2.17 66.33 ± 9.238 4.85 × 10 −11
8 47.5 ± 2.01 68.33 ± 6.351 9.10 × 10 −14 47.9 ± 0.674 71 7.90 × 10 −14
9 67.8 ± 1.89 70.33 ± 6.351 7.00 × 10 −15 68.6 ± 1.28 66.67 ± 3.512 3.40 × 10 −14
10 70.1 ± 7 74 ± 11 4.00 × 10 −14 68.5 ± 4.28 75 ± 6.928 1.55 × 10 −13
11 78.3 ± 9.35 136.3 ± 38.63 9.10 × 10 −12 68.2 ± 1.87 95.67 ± 9.238 4.25 × 10 −12
12 69.3 ± 3.84 99.67 ± 16.80 4.75 × 10 −12 67.3 ± 0.601 95.67 ± 9.238 7.21 × 10 −12
13 118 ± 13.8 131.3 ± 31.75 6.98 × 10 −9 114 ± 1.46 129 1.86 × 10 −9
14 109 ± 2.08 116.7 ± 6.351 1.82 × 10 −11 106 ± 0.780 113 3.21 × 10 −12
15 245 ± 5.95 175.7 ± 5.680 2.19 × 10 −11 246 ± 3.85 177.3 ± 8.262 7.35 × 10 −12
16 252 ± 5.16 175.7 ± 8.981 1.74 × 10 −12 253 ± 4.83 180 ± 8.764 2.00 × 10 −12
17 296 ± 0.901 265 4.36 × 10 −7 292 ± 0.739 265 1.96 × 10 −7
18 307 ± 1.40 265 9.65 × 10 −8 304 ± 2.56 265 2.28 × 10 −8
19 614 ± 15.4 365 9.38 × 10 −9 614 ± 10.2 365 4.35 × 10 −9
20 615 ± 10.1 365 2.39 × 10 −9 613 ± 10 365 2.66 × 10 −10
21 2580 ± 636 702.7 ± 63.51 2.10 × 10 −5 2160 ± 29.5 666 5.64 × 10 −6
22 2150 ± 44.3 666 2.19 × 10 −6 2130 ± 33.2 666 4.96 × 10 −7
7.5 Numerical experiments with JDCOMM software
In this section a Matlab implementation of the JDCOMM method, based on the pseudocode
of Algorithm 3, is used to compute the global minimum of several polynomials.