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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 vi be a right eigenvector corresponding to : . Let and be the corresponding computed eigenvalues and eigenvectors, computed by expert driver routine xGEEVX (see subsection 2.3.4).

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 true eigenvectors vi, 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 example4.11, if and

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

i |  |  | `EERRBD`(i) | true | `VERRBD`(i) | true
1 | 50 | 50.00 |  |  |  |
2 | 2 | 1.899 |  |  |  |
3 | 1 | 1.101 |  |  |  |

- Further Details: Error Bounds for the Nonsymmetric Eigenproblem
  - Overview
  - Balancing and Conditioning
  - Computing s and

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