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