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## 1Purpose

f01ej is the AD Library version of the primal routine f01ejf. Based (in the C++ interface) on overload resolution, f01ej can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine f01ej_AD_f ( n, a, lda, imnorm, ifail)
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail ADTYPE, Intent (Inout) :: a(lda,*) ADTYPE, Intent (Out) :: imnorm 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 f01ej ( handle_t &ad_handle, const Integer &n, ADTYPE a[], const Integer &lda, ADTYPE &imnorm, 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

f01ej is the AD Library version of the primal routine f01ejf.
f01ejf computes the principal matrix logarithm, $\mathrm{log}\left(A\right)$, of a real $n×n$ matrix $A$, with no eigenvalues on the closed negative real line. For further information see Section 3 in the documentation for f01ejf.

## 4References

Al–Mohy A H and Higham N J (2011) Improved inverse scaling and squaring algorithms for the matrix logarithm SIAM J. Sci. Comput. 34(4) C152–C169
Al–Mohy A H, Higham N J and Relton S D (2012) Computing the Fréchet derivative of the matrix logarithm and estimating the condition number SIAM J. Sci. Comput. 35(4) C394–C410
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f01ej 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: n – Integer Input
3: a(lda, $*$) – ADTYPE array Input/Output
4: lda – Integer Input
5: Output
6: ifail – Integer Input/Output

## 6Error Indicators and Warnings

f01ej preserves all error codes from f01ejf and in addition can 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

f01ej is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f01ejf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example finds the principal matrix logarithm of the matrix
 $A = ( 3 −3 1 1 2 1 −2 1 1 1 3 −1 2 0 2 0 ) .$

Language Source File Data Results
Fortran f01ej_a1w_fe.f90 f01ej_a1w_fe.d f01ej_a1w_fe.r
C++ f01ej_a1w_hcppe.cpp f01ej_a1w_hcppe.d f01ej_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran f01ej_t1w_fe.f90 f01ej_t1w_fe.d f01ej_t1w_fe.r
C++ f01ej_t1w_hcppe.cpp f01ej_t1w_hcppe.d f01ej_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f01ej_p0w_fe.f90 f01ej_p0w_fe.d f01ej_p0w_fe.r
C++ f01ej_p0w_hcppe.cpp f01ej_p0w_hcppe.d f01ej_p0w_hcppe.r