F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08AKF (DORMLQ)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08AKF (DORMLQ) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from an $LQ$ factorization computed by F08AHF (DGELQF).

## 2  Specification

 SUBROUTINE F08AKF ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
 INTEGER M, N, K, LDA, LDC, LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), TAU(*), C(LDC,*), WORK(max(1,LWORK)) CHARACTER(1) SIDE, TRANS
The routine may be called by its LAPACK name dormlq.

## 3  Description

F08AKF (DORMLQ) is intended to be used after a call to F08AHF (DGELQF), which performs an $LQ$ 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).

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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:     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:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{M}}\ge 0$.
4:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     K – INTEGERInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
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:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08AHF (DGELQF).
7:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08AKF (DORMLQ) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{K}}\right)$.
8:     TAU($*$) – 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: further details of the elementary reflectors, as returned by F08AHF (DGELQF).
9:     C(LDC,$*$) – 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.
10:   LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08AKF (DORMLQ) is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – 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.
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08AKF (DORMLQ) 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, ${\mathbf{LWORK}}\ge {\mathbf{N}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least ${\mathbf{M}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'R'}$, where $\mathit{nb}$ is the optimal block size.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ or ${\mathbf{LWORK}}=-1$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ or ${\mathbf{LWORK}}=-1$.
13:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

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 ${\mathbf{SIDE}}=\text{'R'}$.