Generalized (or Quotient) Singular Value Decomposition

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## Generalized (or Quotient) 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 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:

1. 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 .

2. 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
--------------------------------------------------------
```
Table 2.16: Computational routines for the generalized singular value decomposition

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