# NAG Library Routine Document

## 1Purpose

f08acf (dgemqrt) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from a $QR$ factorization computed by f08abf (dgeqrt).

## 2Specification

Fortran Interface
 Subroutine f08acf ( side, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info)
 Integer, Intent (In) :: m, n, k, nb, ldv, ldt, ldc Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: t(ldt,*) Real (Kind=nag_wp), Intent (Inout) :: v(ldv,*), c(ldc,*), work(*) Character (1), Intent (In) :: side, trans
C Header Interface
#include nagmk26.h
 void f08acf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, const Integer *nb, double v[], const Integer *ldv, const double t[], const Integer *ldt, double c[], const Integer *ldc, double work[], Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by its LAPACK name dgemqrt.

## 3Description

f08acf (dgemqrt) is intended to be used after a call to f08abf (dgeqrt) which performs a $QR$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).
A common application of this routine is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Section 10 in f08abf (dgeqrt).
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{side}$ – Character(1)Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathbf{trans}$ – Character(1)Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
3:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$. Usually ${\mathbf{k}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}_{A},{n}_{A}\right)$ where ${m}_{A}$, ${n}_{A}$ are the dimensions of the matrix $A$ supplied in a previous call to f08abf (dgeqrt).
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
6:     $\mathbf{nb}$ – IntegerInput
On entry: the block size used in the $QR$ factorization performed in a previous call to f08abf (dgeqrt); this value must remain unchanged from that call.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if ${\mathbf{k}}>0$, ${\mathbf{nb}}\le {\mathbf{k}}$.
7:     $\mathbf{v}\left({\mathbf{ldv}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08abf (dgeqrt) in the first $k$ columns of its array argument a.
8:     $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f08acf (dgemqrt) is called.
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{ldv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{ldv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{t}\left({\mathbf{ldt}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: further details of the orthogonal matrix $Q$ as returned by f08abf (dgeqrt). The number of blocks is $b=⌈\frac{k}{{\mathbf{nb}}}⌉$, where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)×{\mathbf{nb}}$. For the $b$ blocks the upper triangular block reflector factors ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$ are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
10:   $\mathbf{ldt}$ – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08acf (dgemqrt) is called.
Constraint: ${\mathbf{ldt}}\ge {\mathbf{nb}}$.
11:   $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
12:   $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08acf (dgemqrt) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
13:   $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least ${\mathbf{n}}×{\mathbf{nb}}$ if ${\mathbf{side}}=\text{'L'}$ and at least ${\mathbf{m}}×{\mathbf{nb}}$ if ${\mathbf{side}}=\text{'R'}$.
14:   $\mathbf{info}$ – 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 result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08acf (dgemqrt) 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.

## 9Further Comments

The total number of floating-point operations is approximately $2nk\left(2m-k\right)$ if ${\mathbf{side}}=\text{'L'}$ and $2mk\left(2n-k\right)$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this routine is f08aqf (zgemqrt).

## 10Example

See Section 10 in f08abf (dgeqrt).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017