The nonsymmetric eigenvalue problem is more complicated than the symmetric eigenvalue problem. In this subsection, we state the simplest bounds and leave the more complicated ones to subsequent subsections.
Let A be an n-by-n nonsymmetric matrix, with eigenvalues
.
Let vi be a right eigenvector
corresponding to
:
.
Let
and
be the corresponding
computed eigenvalues and eigenvectors, computed by expert driver routine
xGEEVX (see subsection 2.3.4).
The approximate error bounds4.10for the computed eigenvalues are
EPSMCH = SLAMCH( 'E' )
* Compute the eigenvalues and eigenvectors of A
* WR contains the real parts of the eigenvalues
* WI contains the real parts of the eigenvalues
* VL contains the left eigenvectors
* VR contains the right eigenvectors
CALL SGEEVX( 'P', 'V', 'V', 'B', N, A, LDA, WR, WI,
$ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
$ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
IF( INFO.GT.0 ) THEN
PRINT *,'SGEEVX did not converge'
ELSE IF ( N.GT.0 ) THEN
DO 10 I = 1, N
EERRBD(I) = EPSMCH*ABNRM/RCONDE(I)
VERRBD(I) = EPSMCH*ABNRM/RCONDV(I)
10 CONTINUE
ENDIF
For example4.11, if
and
| i |
|
EERRBD(i) | true
|
VERRBD(i) | true
|
|
| 1 | 50 | 50.00 |
|
|
|
|
| 2 | 2 | 1.899 |
|
|
|
|
| 3 | 1 | 1.101 |
|
|
|
|