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4.3. AN EXAMPLE 47
Using the matrices A T x i
, one can compute the coordinates of the stationary points
of the polynomial p λ : the eigenvalues of each matrix A T x i
are equal to the values of x i
evaluated at the points of V (I), which are equal to the i th coordinates of the stationary
points of the polynomial p λ . As a consequence, one may also choose the polynomial
r equal to the polynomial p λ (x 1 ,...,x n ), which yields the matrix A T p λ
. The matrix
A T p λ
makes it possible to evaluate p λ at its stationary points: the eigenvalues of A T p λ
are the function values of p λ at the points of V (I), i.e., at the (complex) solutions of
the system of equations d (i) (x 1 ,...,x n )=0fori =1,...,n.
A straightforward deployment of the Stetter-Möller matrix method for computing
the global minimum and an associated global minimizer (over R n ) for the real dominated
polynomial p λ now proceeds as follows. As a suitable choice for the polynomial
r, the polynomial p λ is chosen and the corresponding matrix A T p λ
is constructed.
Then its eigenvalues and corresponding eigenvectors are computed. Note that p λ is
defined as a real polynomial and that therefore only real eigenvalues of the aforementioned
matrix are of interest. The smallest real eigenvalue obtained in this way
yields the global minimum and the corresponding minimizer(s) can be read off from
the eigenvector which is structured in the same way as the basis B in Equation (4.5)
is structured (because of the same reason mentioned in Section 3.3). Note that the
eigenvector is structured if and only if the geometric multiplicity of the smallest real
eigenvalue is one. For higher multiplicities the corresponding eigenspace may be larger
and the eigenvector is not structured (see again Section 3.3).
Remark 4.1. Another approach to retrieve the global minimizer is to apply the
eigenvector v, corresponding to the smallest real eigenvalue of A T p λ
, to the matrices
A T x 1
,...,A T x n
. The eigenvalues ξ 1 ,...,ξ n are obtained by solving ξ 1 from the
equation A T x i
v = ξ i v for i =1,...,n. The point (ξ 1 ,...,ξ n ) found, yields the corresponding
global minimizer. Using this approach the matrices A T x 1
,...,A T x n
have to be
constructed explicitly, which is not necessary and not efficient as the minimizers can
be read off from the eigenvector at hand, as shown previously. But if the eigenvalue
under consideration has a multiplicity larger than one, the corresponding eigenvectors
may not necessarily contain the Stetter structure and this could then be an approach
to compute the minimizers.
Remark 4.2. It is possible that there exists a real eigenvalue of the matrix A T p λ
which
has a non-real global minimizer. Non-real global minimizers which yield a real criterion
value, when plugged into the polynomial p λ , are discarded because they are of
no interest in this application.
4.3 An example
Let the dominated polynomial to be minimized be chosen as:
p 1 (x 1 ,x 2 )=(x 4 1 + x 4 2)+x 3 1 +2x 2 1 +3x 1 x 2 (4.8)
for which n =2,d =2,λ = 1, and q(x 1 ,x 2 )=x 3 1 +2x 2 1 +3x 1 x 2 . This dominated
polynomial is visualized in Figure 4.2. From this figure it is clear that there are