Arguments

A

(input/output) REAL or COMPLEX array, shape $(:,:)$ with $size({\bf A},1) = m$ and $size({\bf A},2) = n$.
On entry, the matrix $A$.
On exit, A contains the triangular matrix $R$, or part of $R$, as follows:
If $m-k-l \geq 0$, then $R$ is stored in ${\bf A}_{1:k+l,n-k-l+1:n}$.
If $m-k-l < 0$, then the matrix

$$\pmatrix{R_{11} & R_{12} & R_{13} \cr 0 & R_{22} & R_{23} }$$

is stored in ${\bf A}_{1:m, n-k-l+1:n}$.

B

(input/output) REAL or COMPLEX array, shape $(:,:)$ with $size({\bf B},1) = p$ and $size({\bf B},2) = n$.
On entry, the matrix $B$.
On exit, if $m-k-l < 0$, then $R_{33}$ is stored in ${\bf B}_{m-k+1:l,n+m-k-l+1:n}$.

ALPHA

(output) REAL array, shape $(:)$ with $size({\bf ALPHA}) = n$
The real scalars $\alpha_i$, $i = 1,2,\dots,k+l$.

BETA

(output) REAL array, shape $(:)$ with $size$(BETA) $= n$.
The real scalars $\beta_i$, $i = 1,2,\dots,k+l$.

Note: The generalized singular values of the pair $(A,\,B)$ are

$$\sigma_i = {\bf ALPHA}_i / {\bf BETA}_i, \;\;\; i = 1, 2, \ldots , k + l.$$

If $k + l < n$, then ALPHA$_{k+l+1:n} =$ BETA $_{k+l+1:n} = 0$.

K, L

Optional (output) INTEGER.
The dimension parameters $k$ and $l$.

U

Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(U,1) $= m$.
The matrix $U$.

V

Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(V,1) $= p$.
The matrix $V$.

Q

Optional (output) REAL or COMPLEX square array, shape $(:,:)$ with $size$(Q,1) $= n$.
The matrix $Q$.

IWORK

Optional (output) INTEGER array, shape$(:)$ with $size({\bf IWORK}) = n$.
IWORK contains sorting information. More precisely, the loop

 		 for $i = k+1, \min(m,k+l)$ 


swap ALPHA$_i$ and ALPHA$_{{\bf IWORK}_i}$ 


end       

will sort ALPHA so that ALPHA$_1 \geq$ ALPHA $_2 \geq \dots \geq$ ALPHA$_n$.

INFO

Optional (output) INTEGER.


If INFO is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20].