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f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zggsvd3 (f08vqc)

## 1  Purpose

nag_zggsvd3 (f08vqc) computes the generalized singular value decomposition (GSVD) of an $m$ by $n$ complex matrix $A$ and a $p$ by $n$ complex matrix $B$.

## 2  Specification

 #include #include
 void nag_zggsvd3 (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer n, Integer p, Integer *k, Integer *l, Complex a[], Integer pda, Complex b[], Integer pdb, double alpha[], double beta[], Complex u[], Integer pdu, Complex v[], Integer pdv, Complex q[], Integer pdq, Integer iwork[], NagError *fail)

## 3  Description

Given an $m$ by $n$ complex matrix $A$ and a $p$ by $n$ complex matrix $B$, the generalized singular value decomposition is given by
 $UH A Q = D1 0 R , VH B Q = D2 0 R ,$
where $U$, $V$ and $Q$ are unitary matrices. Let $l$ be the effective numerical rank of $B$ and $\left(k+l\right)$ be the effective numerical rank of the matrix $\left(\begin{array}{c}A\\ B\end{array}\right)$, then the first $k$ generalized singular values are infinite and the remaining $l$ are finite. $R$ is a $\left(k+l\right)$ by $\left(k+l\right)$ nonsingular upper triangular matrix, ${D}_{1}$ and ${D}_{2}$ are $m$ by $\left(k+l\right)$ and $p$ by $\left(k+l\right)$ ‘diagonal’ matrices structured as follows:
if $m-k-l\ge 0$,
 $D1= klkI0l0Cm-k-l00()$
 $D2= kll0Sp-l00()$
 $0R = n-k-lklk0R11R12l00R22()$
where
 $C = diagαk+1,…,αk+l ,$
 $S = diagβk+1,…,βk+l ,$
and
 $C2 + S2 = I .$
$R$ is stored as a submatrix of $A$ with elements ${R}_{ij}$ stored as ${A}_{i,n-k-l+j}$ on exit.
If $m-k-l<0$,
 $D1= km-kk+l-mkI00m-k0C0()$
 $D2= km-kk+l-mm-k0S0k+l-m00Ip-l000()$
 $0R = n-k-lkm-kk+l-mk0R11R12R13m-k00R22R23k+l-m000R33()$
where
 $C = diagαk+1,…,αm ,$
 $S = diagβk+1,…,βm ,$
and
 $C2 + S2 = I .$
$\left(\begin{array}{ccc}{R}_{11}& {R}_{12}& {R}_{13}\\ 0& {R}_{22}& {R}_{23}\end{array}\right)$ is stored as a submatrix of $A$ with ${R}_{ij}$ stored as ${A}_{i,n-k-l+j}$, and ${R}_{33}$ is stored as a submatrix of $B$ with ${\left({R}_{33}\right)}_{ij}$ stored as ${B}_{m-k+i,n+m-k-l+j}$.
The function computes $C$, $S$, $R$ and, optionally, the unitary transformation matrices $U$, $V$ and $Q$.
In particular, if $B$ is an $n$ by $n$ nonsingular matrix, then the GSVD of $A$ and $B$ implicitly gives the SVD of $A×{B}^{-1}$:
 $A B-1 = U D1 D2-1 VH .$
If $\left(\begin{array}{c}A\\ B\end{array}\right)$ has orthonormal columns, then the GSVD of $A$ and $B$ is also equal to the CS decomposition of $A$ and $B$. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
 $AH Ax=λ BH Bx .$
In some literature, the GSVD of $A$ and $B$ is presented in the form
 $UH A X = 0D1 , VH B X = 0D2 ,$
where $U$ and $V$ are orthogonal and $X$ is nonsingular, and ${D}_{1}$ and ${D}_{2}$ are ‘diagonal’. The former GSVD form can be converted to the latter form by setting
 $X = Q I 0 0 R-1 .$
A two stage process is used to compute the GSVD of the matrix pair $\left(A,B\right)$. The pair is first reduced to upper triangular form by unitary transformations using nag_zggsvp3 (f08vuc). The GSVD of the resulting upper triangular matrix pair is then performed by nag_ztgsja (f08ysc) which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

## 5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{jobu}$Nag_ComputeUTypeInput
On entry: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, the unitary matrix $U$ is computed.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_NotU}$.
3:    $\mathbf{jobv}$Nag_ComputeVTypeInput
On entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, the unitary matrix $V$ is computed.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_NotV}$.
4:    $\mathbf{jobq}$Nag_ComputeQTypeInput
On entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, the unitary matrix $Q$ is computed.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_NotQ}$.
5:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
6:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7:    $\mathbf{p}$IntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
8:    $\mathbf{k}$Integer *Output
9:    $\mathbf{l}$Integer *Output
On exit: k and l specify the dimension of the subblocks $k$ and $l$ as described in Section 3; $\left(k+l\right)$ is the effective numerical rank of $\left(\begin{array}{c}A\\ B\end{array}\right)$.
10:  $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ matrix $A$.
On exit: contains the triangular matrix $R$, or part of $R$. See Section 3 for details.
11:  $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12:  $\mathbf{b}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $p$ by $n$ matrix $B$.
On exit: contains the triangular matrix $R$ if $m-k-l<0$. See Section 3 for details.
13:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14:  $\mathbf{alpha}\left[{\mathbf{n}}\right]$doubleOutput
On exit: see the description of beta.
15:  $\mathbf{beta}\left[{\mathbf{n}}\right]$doubleOutput
On exit: alpha and beta contain the generalized singular value pairs of $A$ and $B$, ${\alpha }_{i}$ and ${\beta }_{i}$;
• ${\mathbf{ALPHA}}\left(1:{\mathbf{k}}\right)=1$,
• ${\mathbf{BETA}}\left(1:{\mathbf{k}}\right)=0$,
and if $m-k-l\ge 0$,
• ${\mathbf{ALPHA}}\left({\mathbf{k}}+1:{\mathbf{k}}+{\mathbf{l}}\right)=C$,
• ${\mathbf{BETA}}\left({\mathbf{k}}+1:{\mathbf{k}}+{\mathbf{l}}\right)=S$,
or if $m-k-l<0$,
• ${\mathbf{ALPHA}}\left({\mathbf{k}}+1:{\mathbf{m}}\right)=C$,
• ${\mathbf{ALPHA}}\left({\mathbf{m}}+1:{\mathbf{k}}+{\mathbf{l}}\right)=0$,
• ${\mathbf{BETA}}\left({\mathbf{k}}+1:{\mathbf{m}}\right)=S$,
• ${\mathbf{BETA}}\left({\mathbf{m}}+1:{\mathbf{k}}+{\mathbf{l}}\right)=1$, and
• ${\mathbf{ALPHA}}\left({\mathbf{k}}+{\mathbf{l}}+1:{\mathbf{n}}\right)=0$,
• ${\mathbf{BETA}}\left({\mathbf{k}}+{\mathbf{l}}+1:{\mathbf{n}}\right)=0$.
The notation ${\mathbf{ALPHA}}\left({\mathbf{k}}:{\mathbf{n}}\right)$ above refers to consecutive elements ${\mathbf{alpha}}\left[\mathit{i}-1\right]$, for $\mathit{i}={\mathbf{k}},\dots ,{\mathbf{n}}$.
16:  $\mathbf{u}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{m}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_AllU}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $U$ is stored in
• ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{pdu}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{u}}\left[\left(i-1\right)×{\mathbf{pdu}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u contains the $m$ by $m$ unitary matrix $U$.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, u is not referenced.
17:  $\mathbf{pdu}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
18:  $\mathbf{v}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array v must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdv}}×{\mathbf{p}}\right)$ when ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $V$ is stored in
• ${\mathbf{v}}\left[\left(j-1\right)×{\mathbf{pdv}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{v}}\left[\left(i-1\right)×{\mathbf{pdv}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v contains the $p$ by $p$ unitary matrix $V$.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, v is not referenced.
19:  $\mathbf{pdv}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
• if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{pdv}}\ge 1$.
20:  $\mathbf{q}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q contains the $n$ by $n$ unitary matrix $Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, q is not referenced.
21:  $\mathbf{pdq}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$.
22:  $\mathbf{iwork}\left[{\mathbf{n}}\right]$IntegerOutput
On exit: stores the sorting information. More precisely, if $I$ is the ordered set of indices of alpha containing $C$ (denote as ${\mathbf{alpha}}\left[I\right]$, see beta), then the corresponding elements ${\mathbf{iwork}}\left[I\right]-1$ contain the swap pivots, $J$, that sorts $I$ such that ${\mathbf{alpha}}\left[I\right]$ is in descending numerical order.
The following pseudocode sorts the set $I$:
$\begin{array}{l}\text{for ​}i\in I\\ \phantom{\rule{2em}{0ex}}j={J}_{i}\\ \phantom{\rule{2em}{0ex}}\text{swap ​}{I}_{i}\text{​ and ​}{I}_{j}\\ \text{end}\end{array}$
23:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The Jacobi-type procedure failed to converge.
NE_ENUM_INT_2
On entry, ${\mathbf{jobq}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{jobu}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobv}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
otherwise ${\mathbf{pdv}}\ge 1$.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7  Accuracy

The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E2 = Oε A2 ​ and ​ F2 = Oε B2 ,$
and $\epsilon$ is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

## 8  Parallelism and Performance

nag_zggsvd3 (f08vqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zggsvd3 (f08vqc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

This function replaces the deprecated function nag_zggsvd (f08vnc) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS functions.
The diagonal elements of the matrix $R$ are real.
The real analogue of this function is nag_dggsvd3 (f08vcc).

## 10  Example

This example finds the generalized singular value decomposition
 $A = U Σ1 0R QH , B = V Σ2 0R QH ,$
where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i$
and
 $B = 1 0 -1 0 0 1 0 -1 ,$
together with estimates for the condition number of $R$ and the error bound for the computed generalized singular values.
The example program assumes that $m\ge n$, and would need slight modification if this is not the case.

### 10.1  Program Text

Program Text (f08vqce.c)

### 10.2  Program Data

Program Data (f08vqce.d)

### 10.3  Program Results

Program Results (f08vqce.r)