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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 2.3.5. 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 A12 and B13 are nonsingular upper triangular, and A23 is upper triangular. If m-k-l < 0, the bottom zero block of U1T A Q1 does not appear, and A23 is upper trapezoidal. U1, V1 and Q1 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 A23 and B13 is computed using xTGSJA2.2:

  Here U2, V2 and Q2 are orthogonal (or unitary) matrices, C and S are both real nonnegative diagonal matrices satisfying C2 + S2 = I, S is nonsingular, and R is upper triangular and nonsingular.

Table 2.16: Computational routines for the generalized singular value decomposition
Operation | Single precision | Double precision
| real | complex | real | complex
triangular reduction of A and B | SGGSVP | CGGSVP | DGGSVP | ZGGSVP
GSVD of a pair of triangular matrices | STGSJA | CTGSJA | DTGSJA | ZTGSJA

The reduction to triangular form, performed by xGGSVP, uses QR decomposition with column pivoting for numerical rank determination. See [8] for details.

The generalized singular value decomposition of two triangular matrices, performed by xTGSJA, is done using a Jacobi-like method as described in [83,10].

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