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4.3. AN EXAMPLE 49
the Stetter-Möller matrix method as:
⎛
⎞
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 −1 − 3 4
− 3 4
0 0 0 0 0
0 0 0 0 1 0 0 0 0
A T x 1
=
0 0 0 0 0 1 0 0 0
0 0 0 0 −1 − 3 4
− 3 4
0 0
0 0 0 0 0 0 0 1 0
⎜
⎟
⎝ 0 0 0 0 0 0 0 0 1 ⎠
9
0
16
0 0 0 0 0 −1 − 3 4
⎛
⎞
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
A T x 2
=
0 0 0 0 0 0 0 1 0
.
0 0 0 0 0 0 0 0 1
0 − 3 4
0 0 0 0 0 0 0
⎜
⎝ 0 0 − 3 ⎟
4
0 0 0 0 0 0 ⎠
0
3
4
9
16
9
16
0 0 0 0 0
and
(4.10)
Of course, for such small dimensions computations are easy and it is not difficult to
verify that A T x 1
and A T x 2
commute.
To locate the global minimum of p 1 (x 1 ,x 2 ), attention is first focused on the matrices
A T x 1
and A T x 2
. The eigenvalues of these matrices constitute the values of x 1 and x 2 ,
respectively, at the stationary points of the system of equations (4.9). Using a direct
eigenvalue solver, all the eigenvalues of the matrices A T x 1
and A T x 2
are computed. Both
matrices have distinct eigenvalues, which can be paired by matching their common
eigenspaces, yielding 9 solutions. Restricting to real solutions, 3 stationary points
are obtained: (−0.631899, 0.779656), (0.346826, −0.638348), and (0, 0), which can be
classified, respectively, as a global minimizer, a local minimizer and a saddle point.
The corresponding critical values of p 1 (x 1 ,x 2 ) are obtained by plugging the stationary
points into p 1 (x 1 ,x 2 ), yielding the criterion values −0.402778, −0.201376, and 0,
respectively.
The extension of the Stetter-Möller matrix method mentioned in this chapter,
focuses attention on the matrix A T p . The eigenvalues of the matrix 1(x 1,x 2) AT p 1(x 1,x 2)
are the values of p 1 (x 1 ,x 2 ) at the stationary points. Therefore, the smallest real
eigenvalue of this matrix is a main candidate to yield the global minimum of p 1 (x 1 ,x 2 ).
Any non-real stationary point that corresponds to a real function value p 1 (x 1 ,x 2 )is