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7.5. NUMERICAL EXPERIMENTS WITH JDCOMM SOFTWARE 125
Table 7.15: Comparison between performance of JD and JDCOMM
Method MV A p1 MV A xi Flops ×10 8 Time (s) Global minimum
JD 499 0 13.0 48.5 −2892.25
JDCOMM A x1 99 389 3.2 10.2 −2892.25
JDCOMM A x2 109 499 3.6 12.7 −2892.25
JDCOMM A x3 83 213 2.5 6.0 −2892.25
JDCOMM A x4 85 235 2.6 6.6 −2892.25
JDCOMM A x5 81 191 2.4 5.6 −2892.25
maximal dimensions of the search spaces, this fails: the first computed eigenvalue is a
complex eigenvalue. Only the fifth computed eigenvalue turns out to yield the smallest
real eigenvalue of the matrix A p1 . The five first eigenvalues computed by the JD and
JDCOMM methods are: −3346.2 + 651.314i, −3346.2 − 651.314i, −3146.58 + 661.606i,
−3146.58 − 661.606i and finally −2892.25. The coordinates of the global minimizer
with value −2892.25 are x 1 =3.0922, x 2 = −3.0680, x 3 =3.6112, x 4 =3.6562, and
x 5 = −4.0769. The smallest real eigenvalue of the matrix A p1 lies slightly more into
the inside of the spectrum of eigenvalues, enclosed between two complex conjugate
pairs of eigenvalues, as can be seen in the top-left subfigure of Figure 7.9.
Figure 7.9: Eigenvalue spectra of the matrices A p1
and A xi ,i=1,...,5
The computation time and the number of matrix-vector products needed by the JD
and JDCOMM methods for the computation of the first five eigenvalues, are summarized
in Table 7.15. Also in this situation, the JDCOMM methods with the matrices A xi again
need less computation time. When the maximal dimension of the search spaces is
slightly increased from 70 to 80 the JD methods are capable of computing the smallest