**LA_GGSVD** computes the generalized singular values and,
optionally, the transformation matrices from the generalized
singular value decomposition
of a real or complex matrix pair , where
is and is . The of is written

where , and are orthogonal (unitary) matrices of dimensions , and , respectively. Let be the rank of and the rank of the matrix , and let . Then and are and ``diagonal'' matrices, respectively, and is a nonsingular upper triangular matrix. The detailed structure of , and R depends on the sign of as follows:

The case :

where . We define

The case :

where . We define

In both cases the generalized singular values of the pair are the ratios

The first singular values are infinite. The finite singular values are real and nonnegative.

Note: Some important special cases of the are given in Section 2.2.5.3.