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## 1Purpose

f03bf is the AD Library version of the primal routine f03bff. Based (in the C++ interface) on overload resolution, f03bf can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: id ADTYPE, Intent (In) :: a(lda,*) ADTYPE, Intent (Out) :: d 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:
#include <dco.hpp>
namespace nag {
 void f03bf ( void *&ad_handle, const Integer &n, const ADTYPE a[], const Integer &lda, ADTYPE &d, Integer &id, 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

f03bf is the AD Library version of the primal routine f03bff.
f03bff computes the determinant of a real $n×n$ symmetric positive definite matrix $A$. f07fdf must be called first to supply the symmetric matrix $A$ in Cholesky factorized form. The storage (upper or lower triangular) used by f07fdf is not relevant to f03bff since only the diagonal elements of the factorized $A$ are referenced. For further information see Section 3 in the documentation for f03bff.

## 4References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f03bf 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 handle to the AD configuration data object, as created by x10aa.
2: n – Integer Input
3: a(lda, $*$) – ADTYPE array Input
4: lda – Integer Input
5: Output
6: id – Integer Output
7: ifail – Integer Input/Output

## 6Error Indicators and Warnings

f03bf preserves all error codes from f03bff and in addition can return:
${\mathbf{ifail}}=-89$
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
The routine was called using a mode that has not yet been implemented.
${\mathbf{ifail}}=-443$
This check is only made if the overloaded C++ interface is used with arguments not of type double.
${\mathbf{ifail}}=-444$
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 Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

f03bf is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f03bff, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example computes a Cholesky factorization and calculates the determinant of the real symmetric positive definite matrix
 $( 6 7 6 5 7 11 8 7 6 8 11 9 5 7 9 11 ) .$

Language Source File Data Results
Fortran f03bf_a1w_fe.f90 f03bf_a1w_fe.d f03bf_a1w_fe.r
C++ f03bf_a1w_hcppe.cpp f03bf_a1w_hcppe.d f03bf_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran f03bf_t1w_fe.f90 f03bf_t1w_fe.d f03bf_t1w_fe.r
C++ f03bf_t1w_hcppe.cpp f03bf_t1w_hcppe.d f03bf_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f03bf_p0w_fe.f90 f03bf_p0w_fe.d f03bf_p0w_fe.r
C++ f03bf_p0w_hcppe.cpp f03bf_p0w_hcppe.d f03bf_p0w_hcppe.r