Purpose

LA_GGSVD computes the generalized singular values and, optionally, the transformation matrices from the generalized singular value decomposition $(GSVD)$ of a real or complex matrix pair $(A,\,B)$, where $A$ is $m \times n$ and $B$ is $p \times n$. The $GSVD$ of $(A,\,B)$ is written

$$A = U\,\Sigma_1(0,R)Q^H, \hspace{0.50 cm} B = V\,\Sigma_2(0,R)Q^H$$

where $U$, $V$ and $Q$ are orthogonal (unitary) matrices of dimensions $m \times m$, $p \times p$ and $n\times n$, respectively. Let $l$ be the rank of $B$ and $r$ the rank of the $(m+p)\times n$ matrix $\left( \begin{array}{c} A \\ B \end{array} \right)$, and let $k=r-l$. Then $\Sigma_1$ and $\Sigma_2$ are $m \times (k+l)$ and $p \times (k+l)$ ``diagonal'' matrices, respectively, and $R$ is a $(k+l) \times (k+l)$ nonsingular upper triangular matrix. The detailed structure of $\Sigma_1$, $\Sigma_2$ and R depends on the sign of $(m-k-l)$ as follows:
The case $\;m-k-l \geq 0$:

$$\Sigma_1 \;\;=\;\; \bordermatrix{ & k & l \cr \hspace{0.80... ... 0 \cr \hspace{0.80 cm} l & 0 & C \cr m - k -l & 0 & 0 \cr}$$

$$\Sigma_2 \;\;=\;\; \bordermatrix{ & k & l \cr \hspace{0.30 cm} l & 0 & S \cr p -l & 0 & 0 \cr}$$

$$( 0, R ) \;\;=\;\; \bordermatrix{ & n-k-l & k & l \cr k & 0 & R_{11} & R_{12} \cr l & 0 & 0 & R_{22} \cr}$$

where $C^2 + S^2 = I$. We define

$$\alpha_1 = \alpha_2 = \cdots = \alpha_k =1, \; \alpha_{k+i} = c_{ii}, \;\;\; i = 1, 2, \ldots , l$$

$$\beta_1 = \beta_2 = \cdots = \beta_k =0, \; \beta_{k+i} = s_{ii}, \;\;\; i = 1, 2, \ldots , l$$

The case $\;m-k-l < 0$:

$$\Sigma_1 \;\;=\;\; \bordermatrix{ & k & m-k & k+l-m \cr \hspace{0.37 cm} k & I & 0 & 0 \cr m-k & 0 & C & 0 \cr}$$

$$\Sigma_2 \;\;=\;\; \bordermatrix{ & k & m-k & k+l-m \cr \h... ... k+l-m & 0 & 0 & I \cr \hspace{0.37 cm} p-l & 0 & 0 & 0 \cr}$$

$$(0,\, R) \;\;=\;\; \bordermatrix{ & n-k-l & k & m-k & k+l-... ... 0 & 0 & R_{22} & R_{23} \cr k+l-m & 0 & 0 & 0 & R_{33} \cr}$$

where $C^2 + S^2 = I$. We define

$$\alpha_1 = \alpha_2 = \cdots = \alpha_k =1, \; \alpha_{k+i} =... ...\;\;\; \alpha_{m+1} = \alpha_{m+2} = \cdots = \alpha_{k+l} = 0$$

$$\beta_1 = \beta_2 = \cdots = \beta_k =0, \; \beta_{k+i} = s_{... ...k, \;\;\; \beta_{m+1} = \beta_{m+2} = \cdots = \beta_{k+l} = 1$$

In both cases the generalized singular values of the pair $(A,\,B)$ are the ratios

$$\sigma_i = \alpha_i / \beta_i, \;\;\; i = 1, 2, \ldots , k+l$$

The first $k$ singular values are infinite. The finite singular values are real and nonnegative.
LA_GGSVD computes the real (nonnegative) scalars $\alpha_i, \beta_i, \;\: i = 1, 2, \ldots, k+l$, the matrix $R$, and, optionally, the transformation matrices $U$, $V$ and $Q$.

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