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von Mises-Iteration
x
A
( 0)
is real symmetric:
( ) 1 2 3 N
Ax = λx dim A = N λ > λ ≥ λ ≥… ≥ λ
filled with random numbers, expressed as linear combination of e:
x
( 0)
N
= ∑αie
i=
1
i
i
Iteration scheme x
( v) ( v−1)
= Ax
leads to:
⎡
⎤
v
N
⎢
N
( v)
v v
⎛λ
⎞ ⎥
i
x = ∑αλ i i
ei = λ1 ⎢α1e1+ ∑⎜ ⎟ αiei⎥
i= 1 ⎢ i=
2⎝λ1
⎠ ⎥
⎢
⎣
→0forv→∞
⎥⎦
Ursula van Rienen, Universität Rostock, Institut für Allgemeine Elektrotechnik, AG Computational Electrodynamics
As introduction we will study the von Mises method, also called „power iteration“,
to determine a specific solution for an eigenvalue problem of the form
Ax = λx, dim (A) = N.
We assume that the matrix A is real symmetric with eigenvalues
| λ 1 | > | λ 2 | ≥ | λ 3 | ≥ ... ≥ | λ N |.
We search for the (uniquely defined) eigenvalue λ 1 with large absolute value and
the corresponding eigenvector x 1 .
We start with an initial vector x (0) filled with random numbers. Since the
eigenvectors of A build a complete orthogonal system we can represent x (0) as
linear combination of eigenvectors e 1 , ..., e N as displayed above.
Now we use the iteration scheme
x (v) = Ax (v-1)
And get the expression given above.
The iteration converges against the eigenvector e 1 belonging to the eigenvalue λ 1
with larges absolute value.