The **generalized (or quotient) singular value decomposition**
of an `m`-by-`n` matrix `A` and a `p`-by-`n` matrix `B` is described
in section 2.2.5.
The routines described in this section, are used
to compute the decomposition. The computation proceeds in the following
two stages:

- xGGSVP is used to reduce the matrices
`A`and`B`to triangular form:where and are nonsingular upper triangular, and is upper triangular. If

`m`-`k`- 1 < 0, the bottom zero block of does not appear, and is upper trapezoidal. , and are orthogonal matrices (or unitary matrices if`A`and`B`are complex).`l`is the rank of`B`, and`k`+`l`is the rank of . - The generalized singular value decomposition of two
`l`-by-`l`upper triangular matrices and is computed using xTGSJA:Here , and are orthogonal (or unitary) matrices,

`C`and`S`are both real nonnegative diagonal matrices satisfying ,`S`is nonsingular, and`R`is upper triangular and nonsingular.

-------------------------------------------------------- Single precision Double precision Operation real complex real complex -------------------------------------------------------- triangular reduction SGGSVP CGGSVP DGGSVP ZGGSVP of A and B -------------------------------------------------------- GSVD of a pair of STGSJA CTGSJA DTGSJA ZTGSJA triangular matrices --------------------------------------------------------

The reduction to triangular form, performed by xGGSVP, uses QR decomposition with column pivoting for numerical rank determination. See [12] for details.

The generalized singular value decomposition of two triangular matrices, performed by xTGSJA, is done using a Jacobi-like method as described in [10][62].

Tue Nov 29 14:03:33 EST 1994