The generalized (or quotient) singular value decomposition of an m-by-n matrix A and a p-by-n matrix B is the pair of factorizations
where V, V, Q, R, and are defined as follows.
The generalized singular value decomposition is computed by driver routine xGGSVD (see section 18.104.22.168). We will give error bounds for the generalized singular values in the common case where has full rank r = n. Let and be the values of and , respectively, computed by xGGSVD. The approximate error bound for these values is
Note that if is close to zero, then a true generalized singular value can differ greatly in magnitude from the computed generalized singular value , even if SERRBD is close to its minimum .
Here is another way to interpret SERRBD: if we think of and as representing the subspace S consisting of the straight line through the origin with slope , and similarly and representing the subspace , then SERRBD bounds the acute angle between S and . Note that any two lines through the origin with nearly vertical slopes (very large ) are close together in angle. (This is related to the chordal distance in section 4.10.1.)
SERRBD can be computed by the following code fragment, which for simplicity assumes m > = n. (The assumption r = n implies only that p + m > = n. Error bounds can also be computed when p + m > = n > m, with slightly more complicated code.)
EPSMCH = SLAMCH( 'E' ) * Compute generalized singular values of A and B CALL SGGSVD( 'N', 'N', 'N', M, N, P, K, L, A, LDA, B, $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, $ WORK, IWORK, INFO ) * Compute rank of [A',B']' RANK = K+L IF( INFO.GT.0 ) THEN PRINT *,'SGGSVD did not converge' ELSE IF( RANK.LT.N ) THEN PRINT *,'[A**T,B**T]**T not full rank' ELSE IF ( M .GE. N .AND. N .GT. 0 ) THEN * Compute reciprocal condition number RCOND of R CALL STRCON( 'I', 'U', 'N', N, A, LDA, RCOND, WORK, $ IWORK, INFO ) RCOND = MAX( RCOND, EPSMCH ) SERRBD = EPSMCH / RCOND END IF
For example, if ,
then, to 4 decimal places,
, and the true errors are , and .