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 
be a right eigenvector
corresponding to 
: 
.
Let 
and 
be the corresponding
computed eigenvalues and eigenvectors, computed by expert driver routine
xGEEVX (see subsection 2.2.4).
The approximate error bounds for the computed eigenvalues are
The approximate error
bounds
for the computed eigenvectors ,
which bound the acute angles between the computed eigenvectors and true
eigenvectors , are
These bounds can be computed by the following code fragment:
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 example, if
and
then true eigenvalues, approximate eigenvalues, approximate error bounds, and true errors are
