Error Bounds for the Nonsymmetric Eigenproblem

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Error Bounds for the Nonsymmetric Eigenproblem


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
10       CONTINUE

For example, if and

then true eigenvalues, approximate eigenvalues, approximate error bounds, and true errors are

Tue Nov 29 14:03:33 EST 1994