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7.2. COMPUTING THE GLOBAL MINIMUM USING TARGET SELECTION 113
Table 7.4: Polynomial test cases of various complexities
# n 2d Matrix size Global minimum Terms
1 2 8 49 −7.259 11
2 2 8 49 −4.719 7
3 2 10 81 −20.71 19
4 2 10 81 −24.85 19
5 4 4 81 −24.45 13
6 4 4 81 −37.41 17
7 2 12 121 −3.711 × 10 4 22
8 2 12 121 −52.77 23
9 3 6 125 −20.63 21
10 3 6 125 −142.9 19
11 2 14 169 − 6116 36
12 2 14 169 − 1115 33
13 2 16 225 −2.927 × 10 6 46
14 2 16 225 − 2258 40
15 3 8 343 − 2398 34
16 3 8 343 −884.9 42
17 2 24 529 −1.944 × 10 4 78
18 2 24 529 −1.062 × 10 7 88
19 3 10 729 −1.218 × 10 4 57
20 3 10 729 −1.349 × 10 4 72
21 3 12 1331 −8.138 × 10 5 115
22 3 12 1331 −1.166 × 10 6 112
that is used is given in Table 7.5. The heuristic method used here to compute the
matrix-vector products is the least-increments method and the target selection of the
iterative solvers is chosen such that the smallest real eigenvalues are computed first.
Table 7.5: Settings for JD and JDQZ
JD
JDQZ
# Eigenvalues : 1 # Eigenvalues : 1
Tolerance : 5 × 10 −8 · (2d − 1) n Tolerance : 5 × 10 −8 · (2d − 1) n
Imag. threshold : 1 × 10 −8 · (2d − 1) n Imag. threshold : 1 × 10 −8 · (2d − 1) n
Max # iterations : 1000 Max # iterations : 1000
Min dimension : 0.5 · (2d − 1) n Min dimension : 0.5 · (2d − 1) n
Max dimension : 0.7 · (2d − 1) n Max dimension : 0.7 · (2d − 1) n
Start vector : Random Start vector : Random
Preconditioning : None Preconditioning : None
Extraction : Refined Schur Decomposition
: No
Expansion : JD Conjugate pairs : No
Linear Solver : GMRES Linear solver : GMRES
Fix target : 0.01 Test space : 3
Krylov space : Yes Lin solv tol : 0.7, 0.49
Max # LS it. : 10 Max # LS it. : 15
Thick restart : 1 vector Track : 1.0 × 10 −4
The mean results of the 20 global minima computations of all the polynomials in
the test set are given in Table 7.6. This table shows for JD and JDQZ the mean time
and its standard deviation needed to compute the smallest real eigenvalue (denoted
by mean ± standard deviation), the mean required number of matrix-vector product
and its standard deviation, and the error. This error is defined as the mean of the