F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08VGF (DGGSVP3)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08VGF (DGGSVP3) uses orthogonal 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.

## 2  Specification

 SUBROUTINE F08VGF ( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
 INTEGER M, P, N, LDA, LDB, K, L, LDU, LDV, LDQ, IWORK(N), LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), TOLA, TOLB, U(LDU,*), V(LDV,*), Q(LDQ,*), TAU(N), WORK(max(1,LWORK)) CHARACTER(1) JOBU, JOBV, JOBQ
The routine may be called by its LAPACK name dggsvp3.

## 3  Description

F08VGF (DGGSVP3) computes orthogonal 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 ${\left(\begin{array}{cc}{A}^{\mathrm{T}}& {B}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see routine F08YEF (DTGSJA); the two steps are combined in F08VCF (DGGSVD3).

## 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:     $\mathrm{JOBU}$ – CHARACTER(1)Input
On entry: if ${\mathbf{JOBU}}=\text{'U'}$, the orthogonal matrix $U$ is computed.
If ${\mathbf{JOBU}}=\text{'N'}$, $U$ is not computed.
Constraint: ${\mathbf{JOBU}}=\text{'U'}$ or $\text{'N'}$.
2:     $\mathrm{JOBV}$ – CHARACTER(1)Input
On entry: if ${\mathbf{JOBV}}=\text{'V'}$, the orthogonal matrix $V$ is computed.
If ${\mathbf{JOBV}}=\text{'N'}$, $V$ is not computed.
Constraint: ${\mathbf{JOBV}}=\text{'V'}$ or $\text{'N'}$.
3:     $\mathrm{JOBQ}$ – CHARACTER(1)Input
On entry: if ${\mathbf{JOBQ}}=\text{'Q'}$, the orthogonal matrix $Q$ is computed.
If ${\mathbf{JOBQ}}=\text{'N'}$, $Q$ is not computed.
Constraint: ${\mathbf{JOBQ}}=\text{'Q'}$ or $\text{'N'}$.
4:     $\mathrm{M}$ – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
5:     $\mathrm{P}$ – INTEGERInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{P}}\ge 0$.
6:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
7:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – REAL (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:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08VGF (DGGSVP3) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
9:     $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – REAL (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:   $\mathrm{LDB}$ – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08VGF (DGGSVP3) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{P}}\right)$.
11:   $\mathrm{TOLA}$ – REAL (KIND=nag_wp)Input
12:   $\mathrm{TOLB}$ – 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:   $\mathrm{K}$ – INTEGEROutput
14:   $\mathrm{L}$ – 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:   $\mathrm{U}\left({\mathbf{LDU}},*\right)$ – REAL (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 orthogonal matrix $U$.
If ${\mathbf{JOBU}}=\text{'N'}$, U is not referenced.
16:   $\mathrm{LDU}$ – INTEGERInput
On entry: the first dimension of the array U as declared in the (sub)program from which F08VGF (DGGSVP3) 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:   $\mathrm{V}\left({\mathbf{LDV}},*\right)$ – REAL (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 orthogonal matrix $V$.
If ${\mathbf{JOBV}}=\text{'N'}$, V is not referenced.
18:   $\mathrm{LDV}$ – INTEGERInput
On entry: the first dimension of the array V as declared in the (sub)program from which F08VGF (DGGSVP3) 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:   $\mathrm{Q}\left({\mathbf{LDQ}},*\right)$ – REAL (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 orthogonal matrix $Q$.
If ${\mathbf{JOBQ}}=\text{'N'}$, Q is not referenced.
20:   $\mathrm{LDQ}$ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08VGF (DGGSVP3) 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:   $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
22:   $\mathrm{TAU}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
23:   $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)\right)$ – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
24:   $\mathrm{LWORK}$ – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)routine from which F08VGF (DGGSVP3) 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 ${\mathbf{JOBV}}=\text{'V'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2*{\mathbf{N}}+1,{\mathbf{P}},{\mathbf{M}}\right)$;
• if ${\mathbf{JOBV}}=\text{'N'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2*{\mathbf{N}}+1,{\mathbf{M}}\right)$.
25:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 7  Accuracy

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.

## 8  Parallelism and Performance

F08VGF (DGGSVP3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08VGF (DGGSVP3) 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 F08VEF (DGGSVP) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS routines.
The complex analogue of this routine is F08VUF (ZGGSVP3).

## 10  Example

This example finds the generalized factorization
 $A = UΣ1 0 S QT , B= VΣ2 0 T QT ,$
of the matrix pair $\left(\begin{array}{cc}A& B\end{array}\right)$, where
 $A = 123 321 456 788 and B= -2-33 465 .$

### 10.1  Program Text

Program Text (f08vgfe.f90)

### 10.2  Program Data

Program Data (f08vgfe.d)

### 10.3  Program Results

Program Results (f08vgfe.r)