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## 1Purpose

f08ah is the AD Library version of the primal routine f08ahf (dgelqf). Based (in the C++ interface) on overload resolution, f08ah can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine f08ah_AD_f ( m, n, a, lda, tau, work, lwork, ifail)
 Integer, Intent (In) :: m, n, lda, lwork Integer, Intent (Inout) :: ifail ADTYPE, Intent (Inout) :: a(lda,*), tau(*) ADTYPE, Intent (Out) :: work(max(1,lwork)) Type (c_ptr), Intent (Inout) :: ad_handle
Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:
C++ Interface
#include <dco.hpp>
namespace nag {
 void f08ah ( handle_t &ad_handle, const Integer &m, const Integer &n, ADTYPE a[], const Integer &lda, ADTYPE tau[], ADTYPE work[], const Integer &lwork, Integer &ifail)
}
}
The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types:
double,
dco::ga1s<double>::type,
dco::gt1s<double>::type

## 3Description

f08ah is the AD Library 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 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.
On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object in the NAG AD Library Introduction.
2: m – Integer Input
3: n – Integer Input
4: a(lda, $*$) – ADTYPE array Input/Output
5: lda – Integer Input
6: tau($*$) – ADTYPE array Output
7: work($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)$) – ADTYPE 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 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 Error Handling in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-444$
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f08ah 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.
Description of the primal example.
This example finds the minimum norm solutions of the underdetermined systems of linear equations
 $Ax1= b1 and Ax2= b2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = ( -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 ) and B= ( -2.87 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 ) .$

Language Source File Data Results
Fortran f08ah_a1w_fe.f90 f08ah_a1w_fe.d f08ah_a1w_fe.r
C++ f08ah_a1w_hcppe.cpp f08ah_a1w_hcppe.d f08ah_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran f08ah_t1w_fe.f90 f08ah_t1w_fe.d f08ah_t1w_fe.r
C++ f08ah_t1w_hcppe.cpp f08ah_t1w_hcppe.d f08ah_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f08ah_p0w_fe.f90 f08ah_p0w_fe.d f08ah_p0w_fe.r
C++ f08ah_p0w_hcppe.cpp f08ah_p0w_hcppe.d f08ah_p0w_hcppe.r