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98 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
are given by:
x 1 = qT 1 q2
= −4.030 − 4.016i , where
q1 T q1
⎛
q 1 =
⎜
⎝
1
−4
−13 − 5i
5 + 72i
16 + 140i
507 − 640i
⎞
⎟
⎠
⎛
and q 2 =
⎜
⎝
−4
−5 + 29i
5 + 72i
255 − 283i
507 − 640i
−4613 + 543i
⎞
,
⎟
⎠
(6.46)
and
x 2 = qT 1 q2
= −9.152 − 8.053i , where
q1 T q1
⎛
q 1 =
⎜
⎝
1
−4
−5 + 29i
−13 − 5i
5 + 72i
255 − 283i
⎞
⎟
⎠
⎛
and q 2 =
⎜
⎝
−13 − 5i
5 + 72i
255 − 283i
16 + 140i
507 − 640i
−4613 + 543i
⎞
.
⎟
⎠
(6.47)
The 2-norm of the difference between u and û is 181.349, whereas it was 338.443
using the first projection method of this section. Moreover, the values of x 1 = −4
and x 2 =(−13 − 5i) in the vector u are projected to x 1 =(−4.030 − 4.016i) and
x 2 =(−9.152 − 8.053i) in the projected vector û. These values are closer to the given
values x 1 =(−4 − 4i) and x 2 =(−9 − 8i) than the values computed by the first
projection method.
Although this projection method works for negative and complex entries of the
approximate vectors and it works fast and without the need of additional matrixvector
products with the matrix A, it is not yet implemented in combination with a
Jacobi–Davidson method but it may be addressed in further research.
6.4.2 Embedding Stetter projection in a Jacobi–Davidson method
To embed the projection method of the previous section in a Jacobi–Davidson method,
the Jacobi–Davidson implementation described in [58] is now modified as follows: let
θ j be the approximate smallest real eigenvalue of a matrix A in iteration j of the
iterative eigenvalue solver corresponding to the approximate eigenvector u j . Recall
that the residual r j in iteration j of this eigenvalue pair is defined as r j = Au j −θ j u j .
Now there are two possibilities: