f08 Chapter Contents
f08 Chapter Introduction
NAG C 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 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.

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

## 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 (f08vace.c)

### 9.2  Program Data

Program Data (f08vace.d)

### 9.3  Program Results

Program Results (f08vace.r)