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## 1Purpose

f07fe is the AD Library version of the primal routine f07fef (dpotrs). Based (in the C++ interface) on overload resolution, f07fe can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine f07fe_AD_f ( uplo, n, nrhs, a, lda, b, ldb, ifail)
 Integer, Intent (In) :: n, nrhs, lda, ldb Integer, Intent (Inout) :: ifail ADTYPE, Intent (In) :: a(lda,*) ADTYPE, Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: uplo 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 f07fe ( handle_t &ad_handle, const char *uplo, const Integer &n, const Integer &nrhs, const ADTYPE a[], const Integer &lda, ADTYPE b[], const Integer &ldb, 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

f07fe is the AD Library version of the primal routine f07fef (dpotrs).
f07fef (dpotrs) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by f07fdf. For further information see Section 3 in the documentation for f07fef (dpotrs).

## 4References

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, f07fe 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: uplo – character Input
3: n – Integer Input
4: nrhs – Integer Input
5: a(lda, $*$) – ADTYPE array Input
6: lda – Integer Input
7: b(ldb, $*$) – ADTYPE array Input/Output
8: ldb – 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

f07fe uses the standard NAG ifail mechanism. Any errors indicated via info values returned by f07fef 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

f07fe is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f07fef (dpotrs), modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example solves the system of equations $AX=B$, where
 $A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) and B= ( 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 ) .$
Here $A$ is symmetric positive definite and must first be factorized by f07fd.

Language Source File Data Results
Fortran f07fe_a1w_fe.f90 f07fe_a1w_fe.d f07fe_a1w_fe.r
C++ f07fe_a1w_hcppe.cpp f07fe_a1w_hcppe.d f07fe_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran f07fe_t1w_fe.f90 f07fe_t1w_fe.d f07fe_t1w_fe.r
C++ f07fe_t1w_hcppe.cpp f07fe_t1w_hcppe.d f07fe_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f07fe_p0w_fe.f90 f07fe_p0w_fe.d f07fe_p0w_fe.r
C++ f07fe_p0w_hcppe.cpp f07fe_p0w_hcppe.d f07fe_p0w_hcppe.r