f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dggsvd (f08vac)

## 1  Purpose

nag_dggsvd (f08vac) 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

 #include #include
 void nag_dggsvd (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer n, Integer p, Integer *k, Integer *l, double a[], Integer pda, double b[], Integer pdb, double alpha[], double beta[], double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, Integer iwork[], NagError *fail)

## 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 function 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     jobuNag_ComputeUTypeInput
On entry: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, the orthogonal matrix $U$ is computed.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_NotU}$.
3:     jobvNag_ComputeVTypeInput
On entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, the orthogonal matrix $V$ is computed.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_NotV}$.
4:     jobqNag_ComputeQTypeInput
On entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, the orthogonal matrix $Q$ is computed.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_NotQ}$.
5:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
6:     nIntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7:     pIntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
8:     kInteger *Output
9:     lInteger *Output
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 $\left(\begin{array}{c}A\\ B\end{array}\right)$.
10:   a[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ matrix $A$.
On exit: contains the triangular matrix $R$, or part of $R$. See Section 3 for details.
11:   pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12:   b[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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.
13:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14:   alpha[n]doubleOutput
On exit: see the description of beta.
15:   beta[n]doubleOutput
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 ${\mathbf{alpha}}\left[\mathit{i}-1\right]$, for $\mathit{i}={\mathbf{k}},\dots ,{\mathbf{n}}$.
16:   u[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{m}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_AllU}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $U$ is stored in
• ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{pdu}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{u}}\left[\left(i-1\right)×{\mathbf{pdu}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u contains the $m$ by $m$ orthogonal matrix $U$.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, u is not referenced.
17:   pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
18:   v[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array v must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdv}}×{\mathbf{p}}\right)$ when ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $V$ is stored in
• ${\mathbf{v}}\left[\left(j-1\right)×{\mathbf{pdv}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{v}}\left[\left(i-1\right)×{\mathbf{pdv}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v contains the $p$ by $p$ orthogonal matrix $V$.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, v is not referenced.
19:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
• if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{pdv}}\ge 1$.
20:   q[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q contains the $n$ by $n$ orthogonal matrix $Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, q is not referenced.
21:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$.
22:   iwork[n]IntegerOutput
On exit: stores the sorting information. More precisely, the following loop will sort alpha such that ${\mathbf{alpha}}\left[0\right]\ge {\mathbf{alpha}}\left[1\right]\ge \cdots \ge {\mathbf{alpha}}\left[{\mathbf{n}}-1\right]$.
23:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
The Jacobi-type procedure failed to converge.
NE_ENUM_INT_2
On entry, ${\mathbf{jobq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{jobu}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
otherwise ${\mathbf{pdv}}\ge 1$.
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{jobq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdv}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 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

nag_dggsvd (f08vac) is not threaded by NAG in any implementation.
nag_dggsvd (f08vac) 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.

The complex analogue of this function is nag_zggsvd (f08vnc).

## 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 (f08vace.c)

### 10.2  Program Data

Program Data (f08vace.d)

### 10.3  Program Results

Program Results (f08vace.r)