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 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).
The approximate error
for the computed singular values
The approximate error bounds for the computed singular vectors and , which bound the acute angles between the computed singular vectors and true singular vectors and , 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
then the singular values, approximate error bounds, and true errors are given below.