 
  
  
  
  
 
The singular  value 
decomposition (SVD) of a real m-by-n matrix A is defined as 
follows. Let  . The SVD of A is
. The SVD of A is  (
 
( in the complex case), where 
U and V are orthogonal (unitary) matrices and
 in the complex case), where 
U and V are orthogonal (unitary) matrices and
 is diagonal,
with
 is diagonal,
with  .
The
.
The  are the singular values of A and the leading
r columns
 are the singular values of A and the leading
r columns  of U and
 of U and  of V the 
left and right singular vectors, respectively.
The SVD of a general matrix is computed by PxGESVD 
    
(see subsection 3.2.3).
 of V the 
left and right singular vectors, respectively.
The SVD of a general matrix is computed by PxGESVD 
    
(see subsection 3.2.3).
The approximate error 
bounds
for the computed singular values 
 are
 are

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

These bounds can be computing by the following code fragment:
   
 
      EPSMCH = PSLAMCH( ICTXT, '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 PSGESVD( 'V', 'V', M, N, A, IA, JA, DESCA, S, U, IU, JU,
     $              DESCU, VT, IVT, JVT, DESCVT, WORK, LWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'PSGESVD 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, if  
 and
and

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