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7.1. COMPUTING THE MINIMUM OF A POLYNOMIAL OF ORDER 8 109
given by:
x 1 =+0.876539
x 2 = −0.903966
x 3 =+0.862028
x 4 = −0.835187
(7.3)
(ii) In an attempt to obtain similar information in an independent fashion, the
matrices A xi for all i = 1,...,4, and A p1 have been constructed explicitly using
exact computation. The involved quotient space R[x 1 ,x 2 ,x 3 ,x 4 ]/I is of dimension
(2d − 1) n = 2401. Therefore these matrices have 2401 eigenvalues. A property of
these matrices is that they are highly sparse: the matrix A x1 contains only 4619
non-zero elements (percentage filled: 0.08%), whereas the matrix A p1 contains 43178
non-zero elements (percentage filled: 0.75%). See Figure 7.1 for a representation of
the sparsity structure of the matrices A x1 and A p1 .
Figure 7.1: Sparsity structure of the matrices A x1
and A p1
Building the matrix A p1 took 465 seconds (whereas building the matrices A x1 ,
A x2 , A x3 and A x4 took 56, 60, 63, and 61 seconds, respectively). All the eigenvalues
of the matrix A p1 are computed numerically in two ways: using the eigenvalue solver
Eigensystem in Mathematica and the eigenvalue solver Eig in Matlab. In Table 7.1
the results of these computations are collected.
Table 7.1: Minimal real eigenvalue of the explicit matrix A p1
Method Eigenvalue Time (s)
NSolve solution of system (7.2) 4.09516474435915 744
Eigensystem construct A p1 , compute all eigenvalues 4.09516474435915 465 + 315
Eig construct A p1 , compute all eigenvalues 4.09516474435924 465 + 176
The outcomes of these computations agree up to a limited number of decimal digits
(the fact that the outcomes of the NSolve and the Eigenvalue function in Mathematica