# NAG Library Routine Document

## 1Purpose

f08cjf (dorgrq) generates all or part of the real $n$ by $n$ orthogonal matrix $Q$ from an $RQ$ factorization computed by f08chf (dgerqf).

## 2Specification

Fortran Interface
 Subroutine f08cjf ( m, n, k, a, lda, tau, work, info)
 Integer, Intent (In) :: m, n, k, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include nagmk26.h
 void f08cjf_ (const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name dorgrq.

## 3Description

f08cjf (dorgrq) is intended to be used following a call to f08chf (dgerqf), which performs an $RQ$ factorization of a real matrix $A$ and represents the orthogonal matrix $Q$ as a product of $k$ elementary reflectors of order $n$.
This routine may be used to generate $Q$ explicitly as a square matrix, or to form only its trailing rows.
Usually $Q$ is determined from the $RQ$ factorization of a $p$ by $n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by:
Call dorgrq(n,n,p,a,lda,tau,work,lwork,info)
(note that the matrix $A$ must have at least $n$ rows), or its trailing $p$ rows as:
Call dorgrq(p,n,p,a,lda,tau,work,lwork,info)
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus f08chf (dgerqf) followed by f08cjf (dorgrq) can be used to orthogonalize the rows of $A$.
The information returned by f08chf (dgerqf) also yields the $RQ$ factorization of the trailing $k$ rows of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
Call dorgrq(n,n,k,a,lda,tau,work,lwork,info)
or its leading $k$ columns by:
Call dorgrq(k,n,k,a,lda,tau,work,lwork,info)

## 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 http://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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
3:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08chf (dgerqf).
On exit: the $m$ by $n$ matrix $Q$.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08cjf (dorgrq) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{tau}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: ${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by f08chf (dgerqf).
7:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
8:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08cjf (dorgrq) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ or ${\mathbf{lwork}}=-1$.
9:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = O⁡ε$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08cjf (dorgrq) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $m=k$ this becomes $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this routine is f08cwf (zungrq).

## 10Example

This example generates the first four rows of the matrix $Q$ of the $RQ$ factorization of $A$ as returned by f08chf (dgerqf), where
 $A = -0.57 -1.93 2.30 -1.93 0.15 -0.02 -1.28 1.08 0.24 0.64 0.30 1.03 -0.39 -0.31 0.40 -0.66 0.15 -1.43 0.25 -2.14 -0.35 0.08 -2.13 0.50 .$
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1Program Text

Program Text (f08cjfe.f90)

### 10.2Program Data

Program Data (f08cjfe.d)

### 10.3Program Results

Program Results (f08cjfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017