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100 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
AU − θU to solve Equation (6.49), can be used.
When minimizing the norm ∣ ∣ ( AU − θU ) c ∣ ∣ 2 , the deviation of u = Uc from the
2
Stetter structure can be taken into account too, as follows:
∣ ∣ ∣ ( AU − θU ) c ∣ ∣ ∣ ∣ 2 2 + α ∣ ∣ ∣ ∣ ( Uc− û ) ∣ ∣ ∣ ∣ 2 2 . (6.50)
Note that the only unknown in this equation is c. Equation (6.50) can be written as:
∣ ∣ ∣ Bc − v
∣ ∣
∣ ∣
2
2
where
⎛
B =
⎜
⎝
AU − θU
αU
⎞
⎟
⎠
⎛
and v =
⎜
⎝
0
α û
⎞
.
⎟
⎠
(6.51)
The remaining problem that has to be solved in this Refined Rayleigh-Ritz extraction
method for projection, where the deviation from the Stetter structure is taken into
account, is:
ĉ = argmin
c∈C k ,||c||=1
∣ ∣ ∣∣ Bc − v 2
2
(6.52)
involving the tall rectangular matrix B and the vector v.
The problem of computing the minimum of ∣ ∣ Bc ∣ ∣ 2 2 where c ∈ Ck and ||c|| =1
can be solved by using the Courant-Fischer Theorem as mentioned in [72]. Here the
additional vector −v makes the situation more difficult and as a result this theorem
can not be applied directly here. A solution would be to extend this theorem or to
use the method of Lagrange Multipliers [46] as shown below.
If no projection is used the matrix B = AU − θU in Equation (6.52) is square
since α = 0 and also v = 0. Then a singular value decomposition of the matrix
(AU − θU) yields solutions for ĉ as in the original implementation of the Jacobi–
Davidson method.
When the involved matrix B is not square, the problem (6.52) is solved using the
method of Lagrange Multipliers. To apply such a technique the system of equations
Bc − v has to be transformed first into a more convenient form. Applying a singular
value decomposition to the (possibly) complex valued matrix B yields B = WDV ∗
with unitary matrices W and V . The matrix D is a rectangular matrix containing
the singular values d 1 ,...,d k of the matrix B on its main diagonal. The problem