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

f01ej_a1w_f is the adjoint version of the primal routine f01ejf.

## 2Specification

Fortran Interface
 Subroutine f01ej_a1w_f ( ad_handle, n, a, lda, imnorm, ifail)
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (Inout) :: a(lda,*) Type (nagad_a1w_w_rtype), Intent (Out) :: imnorm Type (c_ptr), Intent (Inout) :: ad_handle
 void f01ej_a1w_f_ ( void *&ad_handle, const Integer &n, nagad_a1w_w_rtype a[], const Integer &lda, nagad_a1w_w_rtype &imnorm, Integer &ifail)
The routine may be called by the names f01ej_a1w_f or nagf_matop_real_gen_matrix_log_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

f01ej_a1w_f is the adjoint 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_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: n – Integer Input
3: a(lda, $*$) – Type (nagad_a1w_w_rtype) array Input/Output
4: lda – Integer Input
5: Output
6: ifail – Integer Input/Output

## 6Error Indicators and Warnings

f01ej_a1w_f preserves all error codes from f01ejf and in addition can 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

f01ej_a1w_f 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.