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88 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
can potentially cause numerical difficulties. To overcome this problem the (natural)
logarithmic transformation is applied to the entries of the eigenvector.
Example 6.1. For n = 2 and d = 2 (hence m = 3) an eigenvector v is structured as
(1, x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2) T . The logarithmic transformation applied
to each of the entries of such a vector v with Stetter structure results in:
⎛
log
⎜
⎝
⎞ ⎛
1
x 1
x 2 1
x 2
= log(x 1 )
⎟ ⎜
⎠ ⎝
x 1 x 2
x 2 1x 2
x 2 2
x 1 x 2 2
x 2 1x 2 2
0
1
2
0
1
2
0
1
2
⎞ ⎛
+ log(x 2 )
⎟ ⎜
⎠ ⎝
0
0
0
1
1
1
2
2
2
⎞
. (6.11)
⎟
⎠
In general a Stetter vector v (where the first entry is normalized to 1) can be
decomposed as follows using the same logarithmic transformation:
⎛
log(v) =
⎜
⎝
log(v 1 )
.
.
log(v N )
⎞
⎟
⎠ = log(x 1)w 1 + log(x 2 )w 2 + ...+ log(x n )w n (6.12)
in which the vectors w 1 ,...,w n are fixed vectors, containing the integer values 0,
1, ..., m− 1 (each m n−1 times) in a specific ordering depending on the monomial
ordering that is used.
Remark 6.1. The right-hand side of Equation (6.12) can be extended without loss of
generality with a term log(x 0 )w 0 , where w 0 =(1,...,1) T , to be able to deal with
eigenvectors in which the first entry is not normalized to 1.
The right-hand side of Equation (6.12) can be used to project a vector that has
an approximate Stetter structure to the desired structure. Thus, we are looking for
complex quantities x 1 ,...,x n such that:
log(û) = log(x 1 )w 1 + log(x 2 )w 2 + ...+ log(x n )w n (6.13)
is a good approximation of log(u).
However, this logarithmic transformation is not straightforward when applying it
to a complex or negative entry in the approximate eigenvector. To overcome this
problem, all entries of a vector u, including complex and negative ones, are written
in polar coordinates. This yields for a general vector z:
log(z) = log(|z|)+i · arg(z), (6.14)