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120 CHAPTER 7. NUMERICAL EXPERIMENTS
Figure 7.4: Eigenvalue spectra of the matrices A p1
and A xi ,i=1,...,4
third columns contain the number of matrix-vector products of the method with
the matrix A p1 and the matrices A xi respectively. The fourth column shows the
amount of floating point operations (flops) needed to perform all the matrix-vector
products; the amount of flops is computed as the sum of the products of the number
of multiplications with the number of non-zeros of the corresponding matrices: i.e.,
the JDCOMM method which iterates with the matrix A x1 in the inner loop and with
the matrix A p1 in the outer loop requires 69 iterations with the matrix A p1 and 459
iterations with the matrix A x1 . This costs 69 × 1326556 + 459 × 63186 = 120534738
flops. Finally, column five shows the required computation time.
It is easy to see that all the methods compute the same leftmost real eigenvalue,
and moreover, that the performance of the JDCOMM method is much better than the
performance of the JD method, in terms of computation time as well as in required
floating point operations.
Figure 7.5, shows the plot of the norms of the residuals in the eigenvalue equation
(at each JD outer step), against the number of matrix-vector products.
To put the performance of the JD approach of this paper into perspective, we
discuss briefly the outcomes of the computation of the global minimum of polynomial
(7.6) by the software packages SOSTOOLS and GloptiPoly, which employ totally different
approaches. Both, SOSTOOLS and GloptiPoly, compute the same global optimum
and locations of polynomial p 1 as shown in Table 7.11 (using the default parameter
settings). To determine these, SOSTOOLS uses 19.3 seconds and GloptiPoly uses 17.2
seconds. This shows that the JDCOMM method has the potential to outperform these