f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_ztgsja (f08ysc)

## 1  Purpose

nag_ztgsja (f08ysc) computes the generalized singular value decomposition (GSVD) of two complex upper trapezoidal matrices $A$ and $B$, where $A$ is an $m$ by $n$ matrix and $B$ is a $p$ by $n$ matrix.
$A$ and $B$ are assumed to be in the form returned by nag_zggsvp (f08vsc).

## 2  Specification

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

## 3  Description

nag_ztgsja (f08ysc) computes the GSVD of the matrices $A$ and $B$ which are assumed to have the form as returned by nag_zggsvp (f08vsc)
 $A= n-k-lklk(0A12A13) l 0 0 A23 m-k-l 0 0 0 , if ​ m-k-l ≥ 0; n-k-lklk(0A12A13) m-k 0 0 A23 , if ​ m-k-l < 0 ; B= n-k-lkll(00B13) p-l 0 0 0 ,$
where the $k$ by $k$ matrix ${A}_{12}$ and the $l$ by $l$ matrix ${B}_{13}$ are nonsingular upper triangular, ${A}_{23}$ is $l$ by $l$ upper triangular if $m-k-l\ge 0$ and is $\left(m-k\right)$ by $l$ upper trapezoidal otherwise.
nag_ztgsja (f08ysc) computes unitary matrices $Q$, $U$ and $V$, diagonal matrices ${D}_{1}$ and ${D}_{2}$, and an upper triangular matrix $R$ such that
 $UHAQ = D1 0 R , VHBQ = D2 0 R .$
Optionally $Q$, $U$ and $V$ may or may not be computed, or they may be premultiplied by matrices ${Q}_{1}$, ${U}_{1}$ and ${V}_{1}$ respectively.
If $\left(m-k-l\right)\ge 0$ then ${D}_{1}$, ${D}_{2}$ and $R$ have the form
 $D1= klk(I0) l 0 C m-k-l 0 0 ,$
 $D2= kll(0S) p-l 0 0 ,$
 $R = klk(R11R12) l 0 R22 ,$
where $C=\mathrm{diag}\left({\alpha }_{k+1},,,\dots ,,,{\alpha }_{k+l}\right)\text{, }S=\mathrm{diag}\left({\beta }_{k+1},,,\dots ,,,{\beta }_{k+l}\right)$.
If $\left(m-k-l\right)<0$ then ${D}_{1}$, ${D}_{2}$ and $R$ have the form
 $D1= km-kk+l-mk(I00) m-k 0 C 0 ,$
 $D2= km-kk+l-mm-k(0S0) k+l-m 0 0 I p-l 0 0 0 ,$
 $R = km-kk+l-mk(R11R12R13) m-k 0 R22 R23 k+l-m 0 0 R33 ,$
where $C=\mathrm{diag}\left({\alpha }_{k+1},,,\dots ,,,{\alpha }_{m}\right)\text{, }S=\mathrm{diag}\left({\beta }_{k+1},,,\dots ,,,{\beta }_{m}\right)$.
In both cases the diagonal matrix $C$ has real non-negative diagonal elements, the diagonal matrix $S$ has real positive diagonal elements, so that $S$ is nonsingular, and ${C}^{2}+{S}^{2}=1$. See Section 2.3.5.3 of Anderson et al. (1999) for further information.
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 ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u must contain a unitary matrix ${U}_{1}$ on entry, and the product ${U}_{1}U$ is returned.
If ${\mathbf{jobu}}=\mathrm{Nag_InitU}$, u is initialized to the unit matrix, and the unitary matrix $U$ is returned.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_InitU}$, $\mathrm{Nag_AllU}$ or $\mathrm{Nag_NotU}$.
3:     jobvNag_ComputeVTypeInput
On entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v must contain a unitary matrix ${V}_{1}$ on entry, and the product ${V}_{1}V$ is returned.
If ${\mathbf{jobv}}=\mathrm{Nag_InitV}$, v is initialized to the unit matrix, and the unitary matrix $V$ is returned.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, $\mathrm{Nag_InitV}$ or $\mathrm{Nag_NotV}$.
4:     jobqNag_ComputeQTypeInput
On entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q must contain a unitary matrix ${Q}_{1}$ on entry, and the product ${Q}_{1}Q$ is returned.
If ${\mathbf{jobq}}=\mathrm{Nag_InitQ}$, q is initialized to the unit matrix, and the unitary matrix $Q$ is returned.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, $\mathrm{Nag_InitQ}$ or $\mathrm{Nag_NotQ}$.
5:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
6:     pIntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
7:     nIntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
8:     kIntegerInput
9:     lIntegerInput
On entry: k and l specify the sizes, $k$ and $l$, of the subblocks of $A$ and $B$, whose GSVD is to be computed by nag_ztgsja (f08ysc).
10:   a[$\mathit{dim}$]ComplexInput/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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ 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: if $m-k-l\ge 0$, ${\mathbf{A}}\left(1:k+l,n-k-l+1:n\right)$ contains the $\left(k+l\right)$ by $\left(k+l\right)$ upper triangular matrix $R$.
If $m-k-l<0$, ${\mathbf{A}}\left(1:m,n-k-l+1:n\right)$ contains the first $m$ rows of the $\left(k+l\right)$ by $\left(k+l\right)$ upper triangular matrix $R$, and the submatrix ${R}_{33}$ is returned in ${\mathbf{B}}\left(m-k+1:l,n+m-k-l+1:n\right)$.
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[$\mathit{dim}$]ComplexInput/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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{B}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ 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: if $m-k-l<0$, ${\mathbf{B}}\left(m-k+1:l,n+m-k-l+1:n\right)$ contains the submatrix ${R}_{33}$ of $R$.
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
15:   tolbdoubleInput
On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by nag_zggsvp (f08vsc), say
 $tola=maxm,nAε, tolb=maxp,nBε,$
where $\epsilon$ is the machine precision.
16:   alpha[n]doubleOutput
On exit: see the description of beta.
17:   beta[n]doubleOutput
On exit: alpha and beta contain the generalized singular value pairs of $A$ and $B$;
• ${\mathbf{alpha}}\left[\mathit{i}\right]=1$, ${\mathbf{beta}}\left[\mathit{i}\right]=0$, for $\mathit{i}=0,1,\dots ,k-1$, and
• if $m-k-l\ge 0$, ${\mathbf{alpha}}\left[\mathit{i}\right]={\alpha }_{\mathit{i}}$, ${\mathbf{beta}}\left[\mathit{i}\right]={\beta }_{\mathit{i}}$, for $\mathit{i}=k,\dots ,k+l-1$, or
• if $m-k-l<0$, ${\mathbf{alpha}}\left[\mathit{i}\right]={\alpha }_{\mathit{i}}$, ${\mathbf{beta}}\left[\mathit{i}\right]={\beta }_{\mathit{i}}$, for $\mathit{i}=k,\dots ,m-1$ and ${\mathbf{alpha}}\left[\mathit{i}\right]=0$, ${\mathbf{beta}}\left[\mathit{i}\right]=1$, for $\mathit{i}=m,\dots ,k+l-1$.
Furthermore, if $k+l, ${\mathbf{alpha}}\left[\mathit{i}\right]={\mathbf{beta}}\left[\mathit{i}\right]=0$, for $\mathit{i}=k+l,\dots ,n-1$.
18:   u[$\mathit{dim}$]ComplexInput/Output
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 ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,×{\mathbf{pdu}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ 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 entry: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u must contain an $m$ by $m$ matrix ${U}_{1}$ (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if ${\mathbf{jobu}}=\mathrm{Nag_InitU}$, u contains the unitary matrix $U$.
If ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u contains the product ${U}_{1}U$.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, u is not referenced.
19:   pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{jobu}}\ne \mathrm{Nag_NotU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
20:   v[$\mathit{dim}$]ComplexInput/Output
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 ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,×{\mathbf{pdv}}\right)$ when ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ 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 entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v must contain an $p$ by $p$ matrix ${V}_{1}$ (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if ${\mathbf{jobv}}=\mathrm{Nag_InitV}$, v contains the unitary matrix $V$.
If ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v contains the product ${V}_{1}V$.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, v is not referenced.
21:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{jobv}}\ne \mathrm{Nag_NotV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{pdv}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{pdv}}\ge 1$.
22:   q[$\mathit{dim}$]ComplexInput/Output
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 ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,×{\mathbf{pdq}}\right)$ when ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ 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 entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q must contain an $n$ by $n$ matrix ${Q}_{1}$ (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if ${\mathbf{jobq}}=\mathrm{Nag_InitQ}$, q contains the unitary matrix $Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q contains the product ${Q}_{1}Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, q is not referenced.
23:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{jobq}}\ne \mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$.
24:   ncycleInteger *Output
On exit: the number of cycles required for convergence.
25:   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 procedure does not converge after $40$ cycles.
NE_ENUM_INT_2
On entry, ${\mathbf{jobq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{jobu}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobu}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobu}}\ne \mathrm{Nag_NotU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
otherwise ${\mathbf{pdv}}\ge 1$.
On entry, ${\mathbf{jobv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobv}}\ne \mathrm{Nag_NotV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
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 ${\mathbf{jobq}}\ne \mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
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
 $E2 = O⁡ε A2 and F2= O⁡ε B2 ,$
and $\epsilon$ is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

## 8  Parallelism and Performance

nag_ztgsja (f08ysc) is not threaded by NAG in any implementation.
nag_ztgsja (f08ysc) 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 real analogue of this function is nag_dtgsja (f08yec).

## 10  Example

This example finds the generalized singular value decomposition
 $A = UΣ1 0 R QH , B= VΣ2 0 R QH ,$
of the matrix pair $\left(A,B\right)$, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i$
and
 $B = 1 0 -1 0 0 1 0 -1 .$

### 10.1  Program Text

Program Text (f08ysce.c)

### 10.2  Program Data

Program Data (f08ysce.d)

### 10.3  Program Results

Program Results (f08ysce.r)