# NAG Library Routine Document

## 1Purpose

f08vuf (zggsvp3) uses unitary transformations to simultaneously reduce the $m$ by $n$ matrix $A$ and the $p$ by $n$ matrix $B$ to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). For sufficiently large problems, a blocked algorithm is used to make best use of level 3 BLAS.

## 2Specification

Fortran Interface
 Subroutine f08vuf ( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, tau, work, info)
 Integer, Intent (In) :: m, p, n, lda, ldb, ldu, ldv, ldq, lwork Integer, Intent (Out) :: k, l, iwork(n), info Real (Kind=nag_wp), Intent (In) :: tola, tolb Real (Kind=nag_wp), Intent (Out) :: 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) :: tau(n), work(max(1,lwork)) Character (1), Intent (In) :: jobu, jobv, jobq
#include nagmk26.h
 void f08vuf_ ( const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *p, const Integer *n, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, const double *tola, const double *tolb, Integer *k, Integer *l, Complex u[], const Integer *ldu, Complex v[], const Integer *ldv, Complex q[], const Integer *ldq, Integer iwork[], double rwork[], Complex tau[], Complex work[], const Integer *lwork, Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by its LAPACK name zggsvp3.

## 3Description

f08vuf (zggsvp3) computes unitary matrices $U$, $V$ and $Q$ such that
where the $k$ by $k$ matrix ${A}_{12}$ and $l$ by $l$ matrix ${B}_{13}$ are nonsingular upper triangular; ${A}_{23}$ is $l$ by $l$ upper triangular if $m-k-l\ge 0$ and is $\left(m-k\right)$ by $l$ upper trapezoidal otherwise. $\left(k+l\right)$ is the effective numerical rank of the $\left(m+p\right)$ by $n$ matrix ${\left(\begin{array}{cc}{A}^{\mathrm{H}}& {B}^{\mathrm{H}}\end{array}\right)}^{\mathrm{H}}$.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see routine f08ysf (ztgsja); the two steps are combined in f08vqf (zggsvd3).

## 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 http://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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{p}$ – IntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
6:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/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 (or trapezoidal) matrix described in Section 3.
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08vuf (zggsvp3) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
9:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/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 described in Section 3.
10:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08vuf (zggsvp3) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
11:   $\mathbf{tola}$ – Real (Kind=nag_wp)Input
12:   $\mathbf{tolb}$ – Real (Kind=nag_wp)Input
On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix $B$ and a subblock of $A$. Generally, they are set to
 $tola=maxm,nAε, tolb=maxp,nBε,$
where $\epsilon$ is the machine precision.
The size of tola and tolb may affect the size of backward errors of the decomposition.
13:   $\mathbf{k}$ – IntegerOutput
14:   $\mathbf{l}$ – IntegerOutput
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 ${\left(\begin{array}{cc}{{\mathbf{a}}}^{\mathrm{T}}& {{\mathbf{b}}}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.
15:   $\mathbf{u}\left({\mathbf{ldu}},*\right)$ – Complex (Kind=nag_wp) arrayOutput
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 unitary matrix $U$.
If ${\mathbf{jobu}}=\text{'N'}$, u is not referenced.
16:   $\mathbf{ldu}$ – IntegerInput
On entry: the first dimension of the array u as declared in the (sub)program from which f08vuf (zggsvp3) 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) arrayOutput
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 unitary matrix $V$.
If ${\mathbf{jobv}}=\text{'N'}$, v is not referenced.
18:   $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f08vuf (zggsvp3) 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) arrayOutput
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 unitary matrix $Q$.
If ${\mathbf{jobq}}=\text{'N'}$, q is not referenced.
20:   $\mathbf{ldq}$ – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which f08vuf (zggsvp3) 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:   $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
22:   $\mathbf{rwork}\left(2×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
23:   $\mathbf{tau}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
24:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
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.
25:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08vuf (zggsvp3) 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 ${\mathbf{jobv}}=\text{'V'}$, ${\mathbf{lwork}}=-1$ or ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{p}},{\mathbf{m}}\right)$;
• if ${\mathbf{jobv}}=\text{'N'}$, ${\mathbf{lwork}}=-1$ or ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{m}}\right)$.
26:   $\mathbf{info}$ – IntegerOutput
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.

## 7Accuracy

The computed factorization is nearly the exact factorization for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E2 = OεA2 and F2= OεB2,$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08vuf (zggsvp3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vuf (zggsvp3) 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 f08vsf (zggsvp) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS routines.
The real analogue of this routine is f08vgf (dggsvp3).

## 10Example

This example finds the generalized factorization
 $A = UΣ1 0 S QH , B= VΣ2 0 T QH ,$
of the matrix pair $\left(\begin{array}{cc}A& B\end{array}\right)$, 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
 $B = 10-10 010-1 .$

### 10.1Program Text

Program Text (f08vufe.f90)

### 10.2Program Data

Program Data (f08vufe.d)

### 10.3Program Results

Program Results (f08vufe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017