link to my thesis

110 CHAPTER 7. NUMERICAL EXPERIMENTS
are exactly the same suggests that the NSolve function uses a matrix method too to
solve the system of equations). To gain insight in the accuracy of the various methods,
all the locations of the global minimum found with the above described methods have
been used as the starting point for a local search method. Using a thirty digit working
precision, which is a user option in the Mathematica/Maple software, the local search
method obtains in all these cases the following coordinates for the global minimizer:
x 1 =+0.876539213106233894587289929758
x 2 = −0.903966282304642050057296045914
x 3 =+0.862027936174326572650513966373
x 4 = −0.835187476756286528192781820247
(7.4)
The corresponding criterion value of this point is computed as
4.095164 744359157279770316678156. These values have been used as the ‘true’ minimizer
and global minimum of the polynomial p 1 (x 1 ,x 2 ,x 3 ,x 4 ) for the purpose of accuracy
analysis of the numerical outcomes of the various computational approaches.
The norm of the difference between this ‘true’ minimizer and the minimizer computed
by the NSolve/Eigensystem and the Eig methods is 7.10543 × 10 −15 and
8.26006 × 10 −14 , respectively.
Following the approach of the previous chapters, we then proceed to determine
the global minimum of the polynomial (7.1) using the nD-system implementation of
the linear operator A p1 to compute only its smallest real eigenvalue with an iterative
eigenvalue solver (instead of working with the explicit matrix A p1 ). The heuristic
method used here is the least-increments method. The coordinates of the global minimum
are computed from the corresponding eigenvector, employing the Stetter vector
structure. For this purpose the iterative eigenvalue solvers JDQR, JDQZ, and Eigs have
been used. JDQR is a normal – and JDQZ is a generalized iterative eigenvalue solver.
Both methods employ Jacobi–Davidson methods coded in Matlab. The method Eigs
is an iterative standard Matlab eigenproblem solver which uses (restarted) Arnoldimethods
through ARPACK. The selection criterion of the iterative solvers used here
is to focus on the eigenvalues with the smallest magnitude first, hoping to find the
smallest real eigenvalue first.
For a fast convergence of these iterative eigenvalue methods, a balancing technique
is used to balance the linear operator, see [27]. Balancing a linear operator or a matrix
M means finding a similarity transform D −1 MD, with D a diagonal matrix, such
that, for each i, row i and column i have (approximately) the same norm. The
algorithms described in [27] help to reduce the norm of the matrix by using methods
of Perron-Frobenius and the direct iterative method. Such a balancing algorithm is
expected to be a good preconditioner for iterative eigenvalue solvers, especially when
the matrix is not available explicitly.
In Table 7.2 the results of these computations are displayed. The columns denote
the methods used, the minimal real eigenvalue computed, the error as the difference
between this eigenvalue and the ‘true’ eigenvalue computed above, the number of