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

f07fd_a1w_f is the adjoint version of the primal routine f07fdf (dpotrf).

## 2Specification

Fortran Interface
 Subroutine f07fd_a1w_f ( ad_handle, uplo, n, a, lda, ifail)
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo Type (c_ptr), Intent (Inout) :: ad_handle
 void f07fd_a1w_f_ ( void *&ad_handle, const char *uplo, const Integer &n, nagad_a1w_w_rtype a[], const Integer &lda, Integer &ifail, const Charlen length_uplo)
The routine may be called by the names f07fd_a1w_f or nagf_lapacklin_dpotrf_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

f07fd_a1w_f is the adjoint version of the primal routine f07fdf (dpotrf).
f07fdf (dpotrf) computes the Cholesky factorization of a real symmetric positive definite matrix. For further information see Section 3 in the documentation for f07fdf (dpotrf).

## 4References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn14.pdf
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f07fd_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: uplo – character Input
3: n – Integer Input
4: a(lda, $*$) – Type (nagad_a1w_w_rtype) array Input/Output
5: lda – Integer Input
6: 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

f07fd_a1w_f uses the standard NAG ifail mechanism. Any errors indicated via info values returned by f07fdf 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

f07fd_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f07fdf (dpotrf), modified to demonstrate calling the NAG AD Library.