# NAG CL Interfacef08cvc (zgerqf)

## 1Purpose

f08cvc computes an RQ factorization of a complex $m$ by $n$ matrix $A$.

## 2Specification

 #include
 void f08cvc (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, Complex tau[], NagError *fail)
The function may be called by the names: f08cvc, nag_lapackeig_zgerqf or nag_zgerqf.

## 3Description

f08cvc forms the $RQ$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. If $m\le n$, the factorization is given by
 $A = 0 R Q ,$
where $R$ is an $m$ by $m$ lower triangular matrix and $Q$ is an $n$ by $n$ unitary matrix. If $m>n$ the factorization is given by
 $A =RQ ,$
where $R$ is an $m$ by $n$ upper trapezoidal matrix and $Q$ is again an $n$ by $n$ unitary matrix. In the case where $m the factorization can be expressed as
 $A = 0 R Q1 Q2 =RQ2 ,$
where ${Q}_{1}$ consists of the first $\left(n-m\right)$ rows of $Q$ and ${Q}_{2}$ the remaining $m$ rows.
The matrix $Q$ is not formed explicitly, but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Section 9).

## 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 https://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: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/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 $m\le n$, the upper triangle of the subarray ${\mathbf{A}}\left(1:m,n-m+1:n\right)$ contains the $m$ by $m$ upper triangular matrix $R$.
If $m\ge n$, the elements on and above the $\left(m-n\right)$th subdiagonal contain the $m$ by $n$ upper trapezoidal matrix $R$; the remaining elements, with the array tau, represent the unitary matrix $Q$ as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
5: $\mathbf{pda}$Integer Input
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{tau}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the scalar factors of the elementary reflectors.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The computed factorization is the exact factorization of a nearby matrix $A+E$, where
 $E2 = O⁡ε A2$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08cvc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$, or $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the unitary matrix $Q$ f08cvc may be followed by a call to f08cwc :
`nag_lapackeig_zungrq(order, n, n, minmn, a, pda, tau, &fail)`
where $\mathtt{minmn}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$, but note that the first dimension of the array a must be at least n, which may be larger than was required by f08cvc. When $m\le n$, it is often only the first $m$ rows of $Q$ that are required and they may be formed by the call:
`nag_lapackeig_zungrq(order, m, n, m, a, pda, tau, c, pdc, &fail)`
To apply $Q$ to an arbitrary $n$ by $p$ real rectangular matrix $C$, f08cvc may be followed by a call to f08cxc . For example:
`nag_lapackeig_zunmrq(Nag_LeftSide, Nag_ConjTrans, n, p, minmn, a, pda, tau, c, pdc, &fail)`
forms the matrix product $C={Q}^{\mathrm{H}}C$.
The real analogue of this function is f08chc.

## 10Example

This example finds the minimum norm solution to the underdetermined equations
 $Ax=b$
where
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i$
and
 $b = -1.35+0.19i 9.41-3.56i -7.57+6.93i .$
The solution is obtained by first obtaining an $RQ$ factorization of the matrix $A$.

### 10.1Program Text

Program Text (f08cvce.c)

### 10.2Program Data

Program Data (f08cvce.d)

### 10.3Program Results

Program Results (f08cvce.r)