Curl-Curl-Eigenvalue Equation - Institut für Allgemeine ...

Sub-Space-Iteration
0
( r )
( r )
( 0)
( 0)
( 0)
i=
1
( λ r)
1 1 i i 1 i
i=
1
n N n
( λ r )
n k i i k i
k= 1 i= 1 k=
1
N
v = v = α e
v = A− I v = α − e
∏
v = A− I v = α − e
=
∑
N
∑
∑ ∏
N
∑
i=
1
i
α P
i
( λ )
e
i n i i
( λ) = ( λ−
)
with polynomial of the order n:
P r
n
n
∏
k = 1
k
Ursula van Rienen, Universität Rostock, Institut für Allgemeine Elektrotechnik, AG Computational Electrodynamics
Thus we need to search for a set of basis vector, i.e. linear independent vectors v (j)
which each can be represented as linear combination of all searched eigenvectors e j
(i
=1…p). (The vectors v (j) themselves need not to be eigenvectors of A.)
Then, an arbitrary vector of this sub-space of p basis vectors does not hold any
component of the remaining (not interesting) eigenvectors e i
with i = p+1, …,
N.
Therefore, our goal in the computation of the basis vectors is to eliminate all
components of these undesired eigenvectors starting with some arbitrary initial vector
v (0) .
This can be reached by repeated multiplication of the initial vector v (0) with the
matrix
A - r k
I (k = 1, …, n).
The spectral shift by r k
can vary now in each iteration step.
Analogous to the procedure in the von Mises-iteration we receive the representation
given above with a polynomial P n
of order n, i.e. the number of iterations.