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:

$$\begin{eqnarray*} U^T_1 A Q_1 & = & \bordermatrix{ & n-k-l & k & l \cr \hfill k... ...-l & k & l \cr \hfill l & 0 & 0 & B_{13} \cr p-l & 0 & 0 & 0 } \end{eqnarray*}$$

where A₁₂ and B₁₃ are nonsingular upper triangular, and A₂₃ is upper triangular. If m-k-l < 0, the bottom zero block of U₁^T 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 $\left( \begin{array}{c} A \\ B \end{array} \right)$.

2.

The generalized singular value decomposition of two l-by-l upper triangular matrices A₂₃ and B₁₃ is computed using xTGSJA^2.2:

$$A_{23} = U_2 C R Q^T_2 \quad \mbox{and} \quad B_{13} = V_2 S R Q^T_2 \; \; .$$

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].