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Error Bounds for the Generalized Singular Value Decomposition

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 U, V, Q, R, and are defined as follows.

- U is m-by-m, V is p-by-p, Q is n-by-n, and all three matrices are orthogonal. If A and B are complex, these matrices are unitary instead of orthogonal, and QT should be replaced by QH in the pair of factorizations.
- R is r-by-r, upper triangular and nonsingular. [0,R] is r-by-n. The integer r is the rank of , and satisfies .
- is m-by-r, is p-by-r, both are real, nonnegative and diagonal, and . Write and , where and lie in the interval from 0 to 1. The ratios are called the generalized singular values of the pair A, B. If , then the generalized singular value is infinite. For details on the structure of , and R, see section 2.3.5.3.

The generalized singular value decomposition is computed by driver routine xGGSVD (see section 2.3.5.3). 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 bound4.10for 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 consisting of the straight line through the origin with slope , and similarly and representing the subspace , then bounds the acute angle between 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 . (The assumption r=n implies only that . Error bounds can also be computed when , 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 example4.11, if ,

then, to 4 decimal places,

, and the true errors are 0, and .

- Further Details: Error Bounds for the Generalized Singular Value Decomposition

Next: Further Details: Error Bounds Up: Accuracy and Stability Previous: Singular Eigenproblems Contents Index