Next: Further Details: Error Bounds Up: Accuracy and Stability Previous: Further Details: Error Bounds Contents Index

#
Error Bounds for the Generalized Nonsymmetric Eigenproblem

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 the form with instead of . The eigenvalue is represented as a pair , where a finite eigenvalue has and an infinite eigenvalue has . As in the standard nonsymmetric eigenvalue problem we have both right and left eigenvectors and , respectively, defined as

| (4.8)

Error bounds for eigenvalues are stated in terms of the distance between pairs and . Let and . Then the chordal distance between and (see section 4.10.1) can equivalently be expressed as the chordal distance between two pairs:

| (4.9)

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 2.3.5.2).

The approximate error bounds4.10for the computed eigenvalues are

The approximate error bounds for the computed eigenvectors , which bound the acute angles between the computed eigenvectors and the true eigenvectors xi are

The same bounds also hold for the computed left eigenvectors.

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 `SLAMCH('E')` and

and

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.

i | 1 | 2 | 3 | 4
| 1 | 2 | 3 | 4
`EERRBD`(i) |  |  |  |
`VERRBD`(i) |  |  |  |

- Further Details: Error Bounds for the Generalized Nonsymmetric Eigenproblem
  - Overview
  - Balancing and Conditioning
  - Computing si, , and ,
  - Singular Eigenproblems

Next: Further Details: Error Bounds Up: Accuracy and Stability Previous: Further Details: Error Bounds Contents Index