We start by stating the simplest error bounds for individual eigenvalues and eigenvectors and leave the more complicated ones to subsequent subsections.
As discussed in section 2.4.8 ,
from a computational point of view it is more natural to define
the generalized nonsymmetric eigenvalue problem in
as a pair
where a finite
and an infinite eigenvalue has .
As in the standard nonsymmetric eigenvalue problem we have both right
and left eigenvectors
respectively, defined as
Now we state our error bounds. Let be the eigenvalues of (A, B), let xi be a right eigenvector corresponding to : and let and be the corresponding eigenvalues and eigenvectors computed by the expert driver routine xGGEVX (see subsection 18.104.22.168).
The approximate error bounds4.10for the computed eigenvalues are
These bounds can be computed by the following code fragment:
EPSMCH = SLAMCH( 'E' ) * Compute the generalized eigenvalues and eigenvectors of (A,B) * ALPHAR/BETA contains the real parts of the eigenvalues * ALPHAI/BETA contains the imaginary parts of the eigenvalues * VL contains the left eigenvectors * VR contains the right eigenvectors CALL SGGEVX( 'P', 'V', 'V', 'B', N, A, LDA, B, LDB, ALPHAR, $ ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, $ RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, $ IWORK, BWORK, INFO ) IF( INFO.GT.0 ) THEN PRINT *,'INFO =', info, ' from SGGEVX.' ELSE IF( N.GT.0 ) THEN ABNORM = SLAPY2( ABNRM, BBNRM ) DO 10 I = 1,N EERRBD(I) = EPSMCH*ABNORM/RCONDE(I) VERRBD(I) = EPSMCH*ABNORM/RCONDV(I) 10 CONTINUE END IF
For example, suppose4.1
For this problem, the exact eigenvalues, eigenvectors, and eigenvalue condition numbers are known. Then the true eigenvalues, computed eigenvalues, approximate error bounds, and true error bounds are given in the following table.