Note: a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype. Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document. Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types. The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.

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## 1Purpose

f08ah_a1w_f is the adjoint version of the primal routine f08ahf (dgelqf).

## 2Specification

Fortran Interface
 Subroutine f08ah_a1w_f ( ad_handle, m, n, a, lda, tau, work, lwork, ifail)
 Integer, Intent (In) :: m, n, lda, lwork Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (Inout) :: a(lda,*), tau(*) Type (nagad_a1w_w_rtype), Intent (Out) :: work(max(1,lwork)) Type (c_ptr), Intent (Inout) :: ad_handle
 void f08ah_a1w_f_ ( void *&ad_handle, const Integer &m, const Integer &n, nagad_a1w_w_rtype a[], const Integer &lda, nagad_a1w_w_rtype tau[], nagad_a1w_w_rtype work[], const Integer &lwork, Integer &ifail)
The routine may be called by the names f08ah_a1w_f or nagf_lapackeig_dgelqf_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

f08ah_a1w_f is the adjoint version of the primal routine f08ahf (dgelqf).
f08ahf (dgelqf) computes the $LQ$ factorization of a real $m×n$ matrix. For further information see Section 3 in the documentation for f08ahf (dgelqf).

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f08ah_a1w_f includes some arguments specific to AD.
A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine. A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.
1: ad_handle – Type (c_ptr) Input/Output
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f.
2: m – Integer Input
3: n – Integer Input
4: a(lda, $*$) – Type (nagad_a1w_w_rtype) array Input/Output
5: lda – Integer Input
6: tau($*$) – Type (nagad_a1w_w_rtype) array Output
7: work($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)$) – Type (nagad_a1w_w_rtype) array Workspace
8: lwork – Integer Input
9: ifail – Integer Input/Output
On entry: must be set to $0$, .
On exit: any errors are indicated as described in Section 6.

## 6Error Indicators and Warnings

f08ah_a1w_f uses the standard NAG ifail mechanism. Any errors indicated via info values returned by f08ahf may be indicated with the same value returned by ifail. In addition, this routine may return:
${\mathbf{ifail}}=-89$
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f08ah_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f08ahf (dgelqf), modified to demonstrate calling the NAG AD Library.