NAG Library Routine Document

1Purpose

f08vaf (dggsvd) computes the generalized singular value decomposition (GSVD) of an $m$ by $n$ real matrix $A$ and a $p$ by $n$ real matrix $B$. f08vaf (dggsvd) is marked as deprecated by LAPACK; the replacement routine is f08vcf (dggsvd3) which makes better use of level 3 BLAS.

2Specification

Fortran Interface
 Subroutine f08vaf ( 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 Integer, Intent (Out) :: k, l, iwork(n), info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), u(ldu,*), v(ldv,*), q(ldq,*) Real (Kind=nag_wp), Intent (Out) :: alpha(n), beta(n), work(max(3*n,m,p)+n) Character (1), Intent (In) :: jobu, jobv, jobq
#include nagmk26.h
 void f08vaf_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *n, const Integer *p, Integer *k, Integer *l, double a[], const Integer *lda, double b[], const Integer *ldb, double alpha[], double beta[], double u[], const Integer *ldu, double v[], const Integer *ldv, double q[], const Integer *ldq, double work[], Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by its LAPACK name dggsvd.

3Description

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 $m-k-l\ge 0$,
 $D1= klkI0l0Cm-k-l00()$
 $D2= kll0Sp-l00()$
 $0R = n-k-lklk0R11R12l00R22()$
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 taking the nonsingular matrix $X$ as
 $X = Q I 0 0 R-1 .$

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

5Arguments

1:     $\mathbf{jobu}$ – 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:     $\mathbf{jobv}$ – 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:     $\mathbf{jobq}$ – 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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{p}$ – IntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
7:     $\mathbf{k}$ – IntegerOutput
8:     $\mathbf{l}$ – 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 $\left(\begin{array}{c}A\\ B\end{array}\right)$.
9:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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:   $\mathbf{lda}$ – 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:   $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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:   $\mathbf{ldb}$ – 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:   $\mathbf{alpha}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: see the description of beta.
14:   $\mathbf{beta}\left({\mathbf{n}}\right)$ – 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:   $\mathbf{u}\left({\mathbf{ldu}},*\right)$ – 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:   $\mathbf{ldu}$ – 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:   $\mathbf{v}\left({\mathbf{ldv}},*\right)$ – 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:   $\mathbf{ldv}$ – 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:   $\mathbf{q}\left({\mathbf{ldq}},*\right)$ – 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:   $\mathbf{ldq}$ – 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:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(3×{\mathbf{n}},{\mathbf{m}},{\mathbf{p}}\right)+{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
22:   $\mathbf{iwork}\left({\mathbf{n}}\right)$ – 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:   $\mathbf{info}$ – 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.
${\mathbf{info}}=1$
If ${\mathbf{info}}=1$, 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

f08vaf (dggsvd) 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.

The complex analogue of this routine is f08vnf (zggsvd).

10Example

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.

10.1Program Text

Program Text (f08vafe.f90)

10.2Program Data

Program Data (f08vafe.d)

10.3Program Results

Program Results (f08vafe.r)

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