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6.4. PROJECTION TO STETTER-STRUCTURE 95
satisfies the following relations:
⎧
⎪⎨
⎪⎩
û 1 = 1
û 2 = û 2
û 3 = û 2 2
û 4 = û 4
û 5 = û 2 · û 4
û 6 = û 2 2 · û 4
û 7 = û 2 4
û 8 = û 2 · û 2 4
û 9 = û 2 2 · û 2 4
(6.42)
Note that there are other possible choices for the values of k =(k 1 ,...,k 9 ) T in Equation
(6.34) which yield the same projected vector û as in (6.41). For example, the
choices k =(0, −2, −2, −1, −2, −1, −2, −1, 0), k =(0, −3, −3, −1, −3, −3, −1, −2, −3)
or k =(0, −2, −1, 0, −1, −1, −1, −2, −1) yield all the same vector û (this is because
for these values the vectors e i· ˆψ in (6.41) are identical).
The 2-norm of the difference between the original vector u and the projected vector
û is 338.443. This norm is larger than the 2-norm of the difference between the original
vector u and the Stetter vector corresponding to (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 , with given values x 1 =(−4 − 4i) and x 2 =(−9 − 8i), which is computed as
13.342. The large norm of u−û is caused by the errors in the last entries of the vector
u − û. These errors are relatively small but have a large influence on the computation
of the norm. That the errors in the last entries of the vector u − û are larger than in
the first entries is explained by the fact that this projection method operates in the
logarithmic domain to regulate the influence of large dominating powers of the Stetter
structure. When looking at the values of x 1 and x 2 before and after the projection we
see that they are projected, respectively, from −4 and (−13 − 5i) in the vector u to
(−3.862 − 3.684i) and (−9.476 − 8.268i) in the projected vector û. The latter values
for x 1 and x 2 are close to the given values of x 1 =(−4 − 4i) and x 2 =(−9 − 8i).
A drawback of the above projection method is that the optimal values for k in
Equation (6.26) are not directly known: there exist various choices for k. Because of
this, it is required to search for the optimal values for k before this projection method
can be applied. This amounts in determining the vector k =(k 1 ,k 2 ,...,k N ) T such
that the current estimate of the eigenpair improves. Currently, no efficient method
to calculate k is implemented other than a brute force search for a fixed maximum
number of iterations. However, this requires a large amount of matrix-vector multiplications,
which is contradictory to the initial goal of projection. To overcome this,
the projection to Stetter structure discussed here is limited to the situation where the
imaginary part of the approximate eigenvector is small enough to be discarded. This
approach is used in the implementations of the Jacobi–Davidson methods and in the
experiments of Chapter 7.