# NAG FL Interfacef08vqf (zggsvd3)

## 1Purpose

f08vqf computes the generalized singular value decomposition (GSVD) of an $m$ by $n$ complex matrix $A$ and a $p$ by $n$ complex matrix $B$.

## 2Specification

Fortran Interface
 Subroutine f08vqf ( jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, beta, u, ldu, v, ldv, q, ldq, work, info)
 Integer, Intent (In) :: m, n, p, lda, ldb, ldu, ldv, ldq, lwork Integer, Intent (Out) :: k, l, iwork(n), info Real (Kind=nag_wp), Intent (Out) :: alpha(n), beta(n), rwork(2*n) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), u(ldu,*), v(ldv,*), q(ldq,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobu, jobv, jobq
#include <nag.h>
 void f08vqf_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *n, const Integer *p, Integer *k, Integer *l, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, double alpha[], double beta[], Complex u[], const Integer *ldu, Complex v[], const Integer *ldv, Complex q[], const Integer *ldq, Complex work[], const Integer *lwork, double rwork[], Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by the names f08vqf, nagf_lapackeig_zggsvd3 or its LAPACK name zggsvd3.

## 3Description

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 routine 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 f08vuf. The GSVD of the resulting upper triangular matrix pair is then performed by f08ysf which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).

## 4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{jobu}$Character(1) Input
On entry: if ${\mathbf{jobu}}=\text{'U'}$, the unitary matrix $U$ is computed.
If ${\mathbf{jobu}}=\text{'N'}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\text{'U'}$ or $\text{'N'}$.
2: $\mathbf{jobv}$Character(1) Input
On entry: if ${\mathbf{jobv}}=\text{'V'}$, the unitary matrix $V$ is computed.
If ${\mathbf{jobv}}=\text{'N'}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\text{'V'}$ or $\text{'N'}$.
3: $\mathbf{jobq}$Character(1) Input
On entry: if ${\mathbf{jobq}}=\text{'Q'}$, the unitary matrix $Q$ is computed.
If ${\mathbf{jobq}}=\text{'N'}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\text{'Q'}$ or $\text{'N'}$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{p}$Integer Input
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
7: $\mathbf{k}$Integer Output
8: $\mathbf{l}$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 $\left(\begin{array}{c}A\\ B\end{array}\right)$.
9: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: contains the triangular matrix $R$, or part of $R$. See Section 3 for details.
10: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08vqf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
11: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
12: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08vqf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
13: $\mathbf{alpha}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: see the description of beta.
14: $\mathbf{beta}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
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 ${\mathbf{alpha}}\left(\mathit{i}\right)$, for $\mathit{i}={\mathbf{k}},\dots ,{\mathbf{n}}$.
15: $\mathbf{u}\left({\mathbf{ldu}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobu}}=\text{'U'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobu}}=\text{'U'}$, u contains the $m$ by $m$ unitary matrix $U$.
If ${\mathbf{jobu}}=\text{'N'}$, u is not referenced.
16: $\mathbf{ldu}$Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08vqf is called.
Constraints:
• if ${\mathbf{jobu}}=\text{'U'}$, ${\mathbf{ldu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{ldu}}\ge 1$.
17: $\mathbf{v}\left({\mathbf{ldv}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$ if ${\mathbf{jobv}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobv}}=\text{'V'}$, v contains the $p$ by $p$ unitary matrix $V$.
If ${\mathbf{jobv}}=\text{'N'}$, v is not referenced.
18: $\mathbf{ldv}$Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08vqf is called.
Constraints:
• if ${\mathbf{jobv}}=\text{'V'}$, ${\mathbf{ldv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{ldv}}\ge 1$.
19: $\mathbf{q}\left({\mathbf{ldq}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobq}}=\text{'Q'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobq}}=\text{'Q'}$, q contains the $n$ by $n$ unitary matrix $Q$.
If ${\mathbf{jobq}}=\text{'N'}$, q is not referenced.
20: $\mathbf{ldq}$Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which f08vqf is called.
Constraints:
• if ${\mathbf{jobq}}=\text{'Q'}$, ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldq}}\ge 1$.
21: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
22: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08vqf is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork must generally be larger than the minimum; increase workspace by, say, $\mathit{nb}×\left({\mathbf{n}}+1\right)$, where $\mathit{nb}$ is the optimal block size.
Constraints:
• if ${\mathbf{jobv}}=\text{'V'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{m}},{\mathbf{p}}\right)$;
• if ${\mathbf{jobv}}=\text{'N'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{m}}\right)$.
23: $\mathbf{rwork}\left(2×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
24: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Output
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 ${\mathbf{iwork}}\left(I\right)$ 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 $I$:
$\begin{array}{l}\text{for ​}i\in I\\ \phantom{\rule{2em}{0ex}}j={J}_{i}\\ \phantom{\rule{2em}{0ex}}\text{swap ​}{I}_{i}\text{​ and ​}{I}_{j}\\ \text{end}\end{array}$
25: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=1$
The Jacobi-type procedure failed to converge.

## 7Accuracy

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.

## 8Parallelism and Performance

f08vqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vqf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

This routine replaces the deprecated routine f08vnf which used an unblocked algorithm and therefore did not make best use of Level 3 BLAS routines.
The diagonal elements of the matrix $R$ are real.
The real analogue of this routine is f08vcf.

## 10Example

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.

### 10.1Program Text

Program Text (f08vqfe.f90)

### 10.2Program Data

Program Data (f08vqfe.d)

### 10.3Program Results

Program Results (f08vqfe.r)