Error Bounds for the Symmetric Eigenproblem

The eigendecomposition of
an ** n**-by-

The approximate error
bounds^{4.10}
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

These bounds can be computed by the following code fragment:

EPSMCH = SLAMCH( 'E' ) * Compute eigenvalues and eigenvectors of A * The eigenvalues are returned in W * The eigenvector matrix Z overwrites A CALL SSYEV( 'V', UPLO, N, A, LDA, W, WORK, LWORK, INFO ) IF( INFO.GT.0 ) THEN PRINT *,'SSYEV did not converge' ELSE IF ( N.GT.0 ) THEN * Compute the norm of A ANORM = MAX( ABS( W(1) ), ABS( W(N) ) ) EERRBD = EPSMCH * ANORM * Compute reciprocal condition numbers for eigenvectors CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO ) DO 10 I = 1, N ZERRBD( I ) = EPSMCH * ( ANORM / RCONDZ( I ) ) 10 CONTINUE ENDIF

For example^{4.11},
if
and

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

i | EERRBD |
true | ZERRBD(i) |
true | |

1 | -.5157 |
||||

2 | .1709 |
||||

3 | 11.34 |