Error Bounds for the Singular Value Decomposition

The singular value decomposition (SVD) of a
real ** m**-by-

The approximate error
bounds^{4.10}for the computed singular values
are

The approximate error bounds for the computed singular vectors and , which bound the acute angles between the computed singular vectors and true singular vectors

These bounds can be computing by the following code fragment.

EPSMCH = SLAMCH( 'E' ) * Compute singular value decomposition of A * The singular values are returned in S * The left singular vectors are returned in U * The transposed right singular vectors are returned in VT CALL SGESVD( 'S', 'S', M, N, A, LDA, S, U, LDU, VT, LDVT, $ WORK, LWORK, INFO ) IF( INFO.GT.0 ) THEN PRINT *,'SGESVD did not converge' ELSE IF ( MIN(M,N) .GT. 0 ) THEN SERRBD = EPSMCH * S(1) * Compute reciprocal condition numbers for singular vectors CALL SDISNA( 'Left', M, N, S, RCONDU, INFO ) CALL SDISNA( 'Right', M, N, S, RCONDV, INFO ) DO 10 I = 1, MIN(M,N) VERRBD( I ) = EPSMCH*( S(1)/RCONDV( I ) ) UERRBD( I ) = EPSMCH*( S(1)/RCONDU( I ) ) 10 CONTINUE END IF

For example^{4.11},
if
and

then the singular values, approximate error bounds, and true errors are given below.

i |
SERRBD |
true | VERRBD()i |
true | UERRBD()i |
true | |

1 | 21.05 | ||||||

2 | 2.370 | ||||||

3 | 1.143 |