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108 CHAPTER 7. NUMERICAL EXPERIMENTS
7.1 An example of computing the global minimum of a polynomial
of order 8
In this section the Stetter-Möller approach, as described in the previous chapters, is
used to compute the value and the location of the global minimum of a Minkowski
dominated polynomial of order 8. The polynomial under consideration here is the
polynomial p λ with d = 4 and m = 7 containing four variables, n = 4. Moreover,
λ equals 1 and the polynomial q in Equation (4.1) is defined as q(x 1 ,x 2 ,x 3 ,x 4 )=
x 1 x 2 x 2 3x 2 4+3x 1 x 2 + x 2 x 3 + x 3 x 2 4+2x 3 x 4 + x 4 +8:
p 1 (x 1 ,x 2 ,x 3 ,x 4 )=(x 8 1 + x 8 2 + x 8 3 + x 8 4)+
x 1 x 2 x 2 3x 2 4 +3x 1 x 2 + x 2 x 3 + x 3 x 2 4 +2x 3 x 4 + x 4 +8.
(7.1)
The first-order conditions of this Minkowski dominated polynomial are given by:
⎧
d (1) (x 1,x 2,x 3,x 4) = x 7 1 + 1 8 x2x2 3x 2 4 + 3 8 x2 = 0
⎪⎨
⎪⎩
d (2) (x 1,x 2,x 3,x 4) = x 7 2 + 1 8 x1x2 3x 2 4 + 3 8 x1 + 1 8 x3 = 0
d (3) (x 1,x 2,x 3,x 4) = x 7 3 + 1 4 x1x2x3x2 4 + 1 8 x2 4 + 1 8 x2 + 1 4 x4 = 0
d (4) (x 1,x 2,x 3,x 4) = x 7 4 + 1 4 x1x2x2 3x 4 + 1 4 x3x4 + 1 4 x3 + 1 8
= 0
(7.2)
Before we discuss the computation of the global minimum of this polynomial using
the Stetter-Möller method with an nD-system in combination with an iterative
eigenvalue solver, we first follow two other, more ‘conventional’, approaches:
(i) the system of first-order partial derivatives which make up the first-order conditions
is solved using standard software techniques. These solutions yield the stationary
points of the polynomial p 1 which are used to compute the critical values.
(ii) the matrix A p1 is constructed explicitly and the global minimum is computed as
the smallest real eigenvalue of this matrix. The eigenvalue computation is performed
using a direct method.
For these two purposes standard routines from the software packages Mathematica
and Matlab have been employed. For all the experiments throughout this section, use
was made of Matlab 7.0.1 and Mathematica 5.2, running on an Intel Pentium PIV
2.8GHz platform with 512 MB of internal memory.
(i) The real solutions of the system of equations (7.2) constitute the stationary
points of p 1 (x 1 ,x 2 ,x 3 ,x 4 ). This system is solved using Mathematica with the NSolve
routine. By substituting each real solution into the polynomial (7.1), we end up
with 11 critical values: 8.003511, 7.329726, 6.742495, 6.723499, 6.045722, 5.866618,
5.731486, 5.624409, 5.491307, 4.482528, and 4.095165. The corresponding stationary
points can be classified as a global minimizer, four local non-global minimizers, and
six saddle points. The smallest one of these critical values yields the global minimum.
The corresponding stationary point of this global minimum with value 4.095165 is