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Error Bounds for the Singular Value Decomposition

The singular value decomposition (SVD) of a real m-by-n matrix A is defined as follows. Let . The SVD of A is ( in the complex case), where U and V are orthogonal (unitary) matrices and is diagonal, with . The are the singular values of A and the leading r columns ui of U and vi of V the left and right singular vectors, respectively. The SVD of a general matrix is computed by xGESVD or xGESDD (see subsection 2.3.4).

The approximate error bounds4.10for 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 vi and ui, are

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 example4.11, 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 |  |  |  |  |  |

- Further Details: Error Bounds for the Singular Value Decomposition

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