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7.5. NUMERICAL EXPERIMENTS WITH JDCOMM SOFTWARE 119
Figure 7.3: Sparsity structure of the matrices A p1
and A xi ,i=1,...,4
Table 7.11: Comparison between performance of JD and JDCOMM
Method MV A p1 MV A xi Flops ×10 8 Time (s) Global minimum
JD 844 0 11.20 50.0 −3.127 × 10 6
JDCOMM A x1 69 459 1.21 5.8 −3.127 × 10 6
JDCOMM A x2 87 657 1.50 7.5 −3.127 × 10 6
JDCOMM A x3 89 679 1.54 7.6 −3.127 × 10 6
JDCOMM A x4 66 426 1.11 4.9 −3.127 × 10 6
−3.127 × 10 6 and is attained at the point x 1 =7.085, x 2 =7.186, x 3 = −6.716, and
x 4 =6.722. The location of these eigenvalues with respect to the whole eigenvalue
spectrum of each matrix is depicted in Figure 7.4.
A more efficient way of computing the global optimum is by using a Jacobi–
Davidson type eigenvalue solver for a set of commuting matrices and try to compute
only the smallest real eigenvalue of our polynomial p 1 . Table 7.11 shows the results
of a conventional JD method and the JDCOMM method for computing the smallest real
eigenvalue of the matrix A p1 , as described in Section 6.5. The JD method used here is
described in [58]. The settings used for both the JD and JDCOMM are: the tolerance on
the residual norm for the convergence of the eigenpair is chosen as 10 −6 ; the restart
parameters for the minimal and maximal dimensions of the search spaces are chosen
as 10 and 30.
The first column of Table 7.11 denotes the method used and the last column shows
the smallest real eigenvalue computed by the corresponding method. The second and