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

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): and 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 and s are not well determined. For example, if then A and B have and , whereas A' and B' have and , which are completely different, even though and . In this case, , so G is nearly rank-deficient.

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.     Next: Error Bounds for Fast Up: Error Bounds for the Previous: Error Bounds for the   Contents   Index
Susan Blackford
1999-10-01