The GSVD algorithm used in LAPACK ([83,10,8]) is backward stable:
Let the computed GSVD of A and B be and . This is nearly the exact GSVD of A+E and B+F in the following sense. E and F are small:
there exist small , , and such that , , and are exactly orthogonal (or unitary):
is the exact GSVD of A+E and B+F. Here p(n) is a modestly growing function of n, and we take p(n)=1 in the above code fragment.
Let and be the square roots of the diagonal entries of the exact and , and let and the square roots of the diagonal entries of the computed and . Let
Then provided G and have full rank n, one can show [96,82] that
In the code fragment we approximate the numerator of the last expression by and approximate the denominator by in order to compute SERRBD; STRCON returns an approximation RCOND to .
We assume that the rank r of G equals n,
because otherwise the
s are not well determined. For example, if
The reason the code fragment assumes is that in this case is stored overwritten on A, and can be passed to STRCON in order to compute RCOND. If , then the first m rows of are stored in A, and the last n-m rows of are stored in B. This complicates the computation of RCOND: either must be copied to a single array before calling STRCON, or else the lower level subroutine SLACON must be used with code capable of solving linear equations with and as coefficient matrices.