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6.4. PROJECTION TO STETTER-STRUCTURE 87
is developed and added as an extension to the Jacobi–Davidson implementation. This
sorting method first sorts the list of approximate complex eigenvalues of the matrix
U ∗ AU in increasing order with respect to the imaginary part. Then all entries with
an imaginary part below a certain threshold are sorted in increasing order of the real
part. With this sorting method the Jacobi–Davidson methods are made to focus on
the desired eigenvalues for this application: the smallest real eigenvalues.
It should be stressed that the use of a threshold for the imaginary part is essential
because of the iterative approach of the solvers. If no thresholding is used then
the solver may not converge to a single eigenvalue. For example, suppose that the
two smallest real eigenvalues are λ 1 = −117.6003 and λ 2 = −93.4018. However,
because of the approximate approach of the eigenvalue solvers these values may arise
as ˆλ 1 = −117.6006+2.316×10 −10 i and ˆλ 2 = −93.4012+4.794×10 −11 i resulting in the
ordering (ˆλ 2 , ˆλ 1 ) T if no thresholding is used. In a next iteration these small imaginary
parts might be interchanged resulting in the opposite ordering (ˆλ 1 , ˆλ 2 ) T . Such a
repeated change of target will have its effect on the convergence of the solver. With
thresholding all imaginary parts below the threshold (i.e., 1 × 10 −10 ) are effectively
treated as zero, thereby reducing this switching of the target.
Numerical experiments using this Jacobi–Davidson implementation are described
in Chapter 7.
6.4 Projection to Stetter-structure
In the global polynomial optimization of a Minkowski dominated polynomial the
eigenvectors of the matrices involved exhibit the Stetter structure as described in
Section 3.3. If an approximate eigenpair is found by an iterative eigenvalue solver,
the corresponding approximate eigenvector will in general not show the precise Stetter
structure. The idea now is to project this vector to a nearby Stetter vector to
accelerate the convergence to a desired accuracy. In this section two such projection
methods are developed together with procedures to integrate this projection step into
the Jacobi–Davidson algorithm.
6.4.1 Projecting an approximate eigenvector
An approximate eigenvector u =(u 1 ,...,u N ) of size N = m n , where m =2d −
1, exhibits the Stetter structure if it is proportional to a vector of the form v =
(v 1 ,...,v N ) T =(1,x 1 ,x 2 1,...,x m−1
1 ,x 2 ,x 1 x 2 ,x 2 1x 2 ,...,x1 m−1 x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2,
..., x m−1
1 x 2 2,...,...,x m−1
1 x m−1
2 ,x 3 ,x 1 x 3 ,...,...,x m−1
1 x m−1
2 ···x m−1
n ) T . See also
Section 4.1. The idea of the projection method in this section is to find values in an
approximate eigenvector u for the quantities x 1 ,x 2 ,...,x n such that the projected
vector of u, the vector û, is close to a nearby Stetter vector v.
However, particularly for polynomials with high total degree, the convergence can
be dominated by higher powers of x 1 ,x 2 ,...,x n if there is an x i with |x i | > 1. On the
other hand, if all x i are of magnitude less than 1, these high powers go to zero. This