f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zggsvp (f08vsc)

## 1  Purpose

nag_zggsvp (f08vsc) uses unitary transformations to simultaneously reduce the $m$ by $n$ matrix $A$ and the $p$ by $n$ matrix $B$ to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD).

## 2  Specification

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

## 3  Description

nag_zggsvp (f08vsc) computes unitary matrices $U$, $V$ and $Q$ such that
where the $k$ by $k$ matrix ${A}_{12}$ and $l$ 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. $\left(k+l\right)$ is the effective numerical rank of the $\left(m+p\right)$ by $n$ matrix ${\left(\begin{array}{cc}{A}^{\mathrm{H}}& {B}^{\mathrm{H}}\end{array}\right)}^{\mathrm{H}}$.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function nag_zggsvd (f08vnc).

## 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 unitary 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 unitary 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 unitary 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:     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:     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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 (or trapezoidal) matrix described in Section 3.
9:     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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 described in Section 3.
11:   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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:   tolbdoubleInput
On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix $B$ and a subblock of $A$. Generally, they are set to
 $tola=maxm,nAε, tolb=maxp,nBε,$
where $\epsilon$ is the machine precision.
The size of tola and tolb may affect the size of backward errors of the decomposition.
14:   kInteger *Output
15:   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 ${\left(\begin{array}{cc}{{\mathbf{a}}}^{\mathrm{T}}& {{\mathbf{b}}}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.
16:   u[$\mathit{dim}$]ComplexOutput
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 unitary 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[$\mathit{dim}$]ComplexOutput
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 unitary 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[$\mathit{dim}$]ComplexOutput
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 unitary 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:   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_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 factorization is nearly the exact factorization for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E2 = OεA2 and F2= OεB2,$
and $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_zggsvp (f08vsc) is not threaded by NAG in any implementation.
nag_zggsvp (f08vsc) 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 Users' Note for your implementation for any additional implementation-specific information.

## 9  Further Comments

The real analogue of this function is nag_dggsvp (f08vec).

## 10  Example

This example finds the generalized factorization
 $A = UΣ1 0 S QH , B= VΣ2 0 T QH ,$
of the matrix pair $\left(\begin{array}{cc}A& B\end{array}\right)$, 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
 $B = 10-10 010-1 .$

### 10.1  Program Text

Program Text (f08vsce.c)

### 10.2  Program Data

Program Data (f08vsce.d)

### 10.3  Program Results

Program Results (f08vsce.r)