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7.5. NUMERICAL EXPERIMENTS WITH JDCOMM SOFTWARE 121
Figure 7.5: Residual norms against MVs with A p1
for the JD and JDCOMM methods
methods in this particular application.
7.5.2 Experiment 2
In a second experiment we consider a slightly larger problem of higher degree: a
polynomial p λ with n =4,d =5,m = 9, and λ = 1 is considered:
p 1 (x 1 ,x 2 ,x 3 ,x 4 )=(x 10
1 + x 10
2 + x 10
3 + x 10
4 )+
−7x 3 1x 4 2x 2 4 +3x 1 x 3 2x 3 3x 2 4 + x 3 2x 4 3x 2 4 +2x 1 x 2 x 6 3x 4 +
−x 4 1x 2 x 3 3 +2x 3 2x 5 4 +3.
(7.7)
The quotient space R[x 1 ,x 2 ,x 3 ,x 4 ]/I has dimension N =(2d−1) n = 6561, which
yields matrices A p1 , A x1 , A x2 , A x3 ,andA x4 of dimensions 6561 × 6561. Table 7.12
shows the differences in the number of non-zero elements of all the involved matrices.
See also Figure 7.6.
Computing all the 6561 eigenvalues of the matrices using a direct method takes
548, 345, 358, 339, and 389 seconds respectively. The results of computing the smallest
real eigenvalue of the matrix A p1 using the JD and JDCOMM methods, are displayed
in Table 7.13. The parameters for the JD methods are slightly changed to 10 −10 for