|
case 1:
a 8*8-matrix
with eigenvalues +/- 10sqrt(10405)
(=1020.049...)
the double eigenvalue 1000 and
two more large eigenvalues 1019... and 1020,
(a dominant cluster), and in addition 0 and
510-100sqrt(26)=0.09805.
direct iteration with shift zero should fail
and with a small negative shift μ should be very slowly
convergent to
10sqrt(10405)+abs(μ)
whereas inverse iteration with shift zero finds
the eigenvector to λ=0 immediately.
|
|
case 2:
[ 4 1 1 ]
[ 2 4 1 ]
[ 0 1 4 ]
with eigenvalues 3,3,6. This matrix is not diagonalizable.
Two linearly independent eigenvectors are [0,1,-1]' and [3,4,2]'.
The power method converges fast, nevertheless.
whereas inverse iteration with zero shift converges very very slowly.
|
| |
case 3:
[ 4 -5 0 3 ]
[ 0 4 -3 -5 ]
[ 5 -3 4 0 ]
[ 3 0 5 4 ]
with eigenvalues 12,1+/- 5i, 2
with the real eigenvectors [1,-1,1,1]' and [1,1,-1,1]'.
Inverse iteration with a large negative shift
which makes 1+/- 5i the absolutely smallest eigenvalue fails,
as well as the power method with shift for example 6.
If convergence is measured by the change in the Rayleighquotient,
then false convergence is observed since this one converges to the real part of the
eigenvalue, hence 1-μ.
|
|
case 4:
A quindiagonal matrix representing a finite difference approximation of the bending of a bar
with diagonals 1,..,1; -4,...,-4;
5,6,..,6,5; -4,...,-4; 1,...1,
and the eigenvalues
16*sin(i*pi/(2(n+1)))**4, i=1,...,n,
where n can be chosen in the range 2,..,30.
choose n
n =
Important : 3 <= n <= 30 !
|