F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08VAF (DGGSVD)

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

F08VAF (DGGSVD) 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 F08VAF ( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO)
 INTEGER M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(max(3*N,M,P)+N) CHARACTER(1) JOBU, JOBV, JOBQ
The routine may be called by its LAPACK name dggsvd.

## 3  Description

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 $\left(k+l\right)$ be the effective numerical rank of the matrix $\left(\begin{array}{c}A\\ B\end{array}\right)$, then $R$ 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 $m-k-l\ge 0$,
 $D1= klk(I0) l 0 C m-k-l 0 0$
 $D2= kll(0S) p-l 0 0$
 $0R = n-k-lklk(0R11R12) l 0 0 R22$
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 $m-k-l<0$,
 $D1= km-kk+l-mk(I00) m-k 0 C 0$
 $D2= km-kk+l-mm-k(0S0) k+l-m 0 0 I p-l 0 0 0$
 $0R = n-k-lkm-kk+l-mk(0R11R12R13) m-k 0 0 R22 R23 k+l-m 0 0 0 R33$
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 taking the nonsingular matrix $X$ as
 $X = Q I 0 0 R-1 .$

## 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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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:     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:     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:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
5:     N – INTEGERInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     P – INTEGERInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{P}}\ge 0$.
7:     K – INTEGEROutput
8:     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:     A(LDA,$*$) – 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:   LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   B(LDB,$*$) – 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:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{P}}\right)$.
13:   ALPHA(N) – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of BETA.
14:   BETA(N) – 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:   U(LDU,$*$) – 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:   LDU – INTEGERInput
On entry: the first dimension of the array U as declared in the (sub)program from which F08VAF (DGGSVD) 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:   V(LDV,$*$) – 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:   LDV – INTEGERInput
On entry: the first dimension of the array V as declared in the (sub)program from which F08VAF (DGGSVD) 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:   Q(LDQ,$*$) – 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:   LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08VAF (DGGSVD) 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:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(3×{\mathbf{N}},{\mathbf{M}},{\mathbf{P}}\right)+{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
22:   IWORK(N) – INTEGER arrayOutput
On exit: stores the sorting information. More precisely, the following loop will sort ALPHA
` for I=K+1, min(M,K+L) swap ALPHA(I) and ALPHA(IWORK(I)) endfor `
such that ${\mathbf{ALPHA}}\left(1\right)\ge {\mathbf{ALPHA}}\left(2\right)\ge \cdots \ge {\mathbf{ALPHA}}\left({\mathbf{N}}\right)$.
23:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}=1$
If ${\mathbf{INFO}}=1$, 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.

The complex analogue of this routine is F08VNF (ZGGSVD).

## 9  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.

### 9.1  Program Text

Program Text (f08vafe.f90)

### 9.2  Program Data

Program Data (f08vafe.d)

### 9.3  Program Results

Program Results (f08vafe.r)