F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08VCF (DGGSVD3)

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

F08VCF (DGGSVD3) computes the generalized singular value decomposition (GSVD) of an $m$ by $n$ real matrix $A$ and a $p$ by $n$ real matrix $B$.

## 2  Specification

 SUBROUTINE F08VCF ( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)
 INTEGER M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, LWORK, IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(max(1,LWORK)) CHARACTER(1) JOBU, JOBV, JOBQ
The routine may be called by its LAPACK name dggsvd3.

## 3  Description

Given an $m$ by $n$ real matrix $A$ and a $p$ by $n$ real matrix $B$, the generalized singular value decomposition is given by
 $UT A Q = D1 0 R , VT B Q = D2 0 R ,$
where $U$, $V$ and $Q$ are orthogonal 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 orthogonal 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 $A{B}^{-1}$:
 $A B-1 = U D1 D2-1 VT .$
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:
 $AT Ax=λ BT Bx .$
In some literature, the GSVD of $A$ and $B$ is presented in the form
 $UT A X = 0D1 , VT B X = 0D2 ,$
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 orthogonal transformations using F08VGF (DGGSVP3). The GSVD of the resulting upper triangular matrix pair is then performed by F08YEF (DTGSJA) 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:     $\mathrm{JOBU}$ – 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:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     $\mathrm{P}$ – INTEGERInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{P}}\ge 0$.
7:     $\mathrm{K}$ – INTEGEROutput
8:     $\mathrm{L}$ – 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 $\left(\begin{array}{c}A\\ B\end{array}\right)$.
9:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – 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 matrix $R$, or part of $R$. See Section 3 for details.
10:   $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08VCF (DGGSVD3) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – 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 $R$ if $m-k-l<0$. See Section 3 for details.
12:   $\mathrm{LDB}$ – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08VCF (DGGSVD3) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{P}}\right)$.
13:   $\mathrm{ALPHA}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of BETA.
14:   $\mathrm{BETA}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput
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:   $\mathrm{U}\left({\mathbf{LDU}},*\right)$ – 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 $m$ by $m$ 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 F08VCF (DGGSVD3) 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 $p$ by $p$ 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 F08VCF (DGGSVD3) 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 $n$ by $n$ 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 F08VCF (DGGSVD3) 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:   $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)\right)$ – 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.
22:   $\mathrm{LWORK}$ – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)routine from which F08VCF (DGGSVD3) 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 ${\mathbf{JOBV}}=\text{'V'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(3*{\mathbf{N}}+1,{\mathbf{M}},{\mathbf{P}}\right)$;
• if ${\mathbf{JOBV}}=\text{'N'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(3*{\mathbf{N}}+1,{\mathbf{M}}\right)$.
23:   $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayOutput
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 $I$:
$\begin{array}{l}\text{for ​}i\in I\\ \phantom{\rule{2em}{0ex}}j={J}_{i}\\ \phantom{\rule{2em}{0ex}}\text{swap ​}{I}_{i}\text{​ and ​}{I}_{j}\\ \text{end}\end{array}$
24:   $\mathrm{INFO}$ – 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.
${\mathbf{INFO}}=1$
The Jacobi-type procedure failed to converge.

## 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

F08VCF (DGGSVD3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08VCF (DGGSVD3) 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.

## 9  Further Comments

This routine replaces the deprecated routine F08VAF (DGGSVD) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS routines.
The complex analogue of this routine is F08VQF (ZGGSVD3).

## 10  Example

This example finds the generalized singular value decomposition
 $A = U Σ1 0R QT , B = V Σ2 0R QT ,$
where
 $A = 1 2 3 3 2 1 4 5 6 7 8 8 and B = -2 -3 3 4 6 5 ,$
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 (f08vcfe.f90)

### 10.2  Program Data

Program Data (f08vcfe.d)

### 10.3  Program Results

Program Results (f08vcfe.r)