The singular value decomposition (SVD) of a real m-by-n matrix A is defined as follows. Let r = min(m , n). The 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 xGESVD (see subsection 2.2.4).
The approximate error bounds 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 and , 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 example, if and
then the singular values, approximate error bounds, and true errors are given below.