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## 1Purpose

c05au is the AD Library version of the primal routine c05auf. Based (in the C++ interface) on overload resolution, c05au can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first and second order.

## 2Specification

Fortran Interface
 Subroutine c05au_AD_f ( ad_handle, x, h, eps, eta, f, a, b, iuser, ruser, ifail)
 Integer, Intent (Inout) :: iuser(*), ifail External :: f ADTYPE, Intent (In) :: h, eps, eta ADTYPE, Intent (Inout) :: x, ruser(*) ADTYPE, Intent (Out) :: a, b 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 {
}
}
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,
dco::gt1s<dco::gt1s<double>::type>::type,
dco::ga1s<dco::gt1s<double>::type>::type,

## 3Description

c05au is the AD Library version of the primal routine c05auf.
c05auf locates a simple zero of a continuous function from a given starting value. It uses a binary search to locate an interval containing a zero of the function, then Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection, to locate the zero precisely. For further information see Section 3 in the documentation for c05auf.

## 4References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

## 5Arguments

In addition to the arguments present in the interface of the primal routine, c05au 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: Input/Output
3: Input
4: Input
5: Input
6: f – Subroutine External Procedure
Note that f is a subroutine in this interface, returning the function value via the additional output parameter retval.
The specification of f is:
Fortran Interface
 Subroutine f ( ad_handle, x, retval, iuser, ruser)
 Integer, Intent (Inout) :: iuser(*) ADTYPE, Intent (In) :: x ADTYPE, Intent (Inout) :: ruser(*) ADTYPE, Intent (Out) :: retval Type (c_ptr), Intent (Inout) :: ad_handle
On entry: a handle to the AD configuration data object.
2: Input
On exit: the value of $f$ at the specified point.
4: iuser – Integer array User Workspace
7: Output
8: Output
9: liuser Input
User workspace dimension (C++ only), see x10af to specify the dimension from Fortran.
10: iuser($*$) – Integer array User Workspace
11: lruser Input
User workspace dimension (C++ only), see x10ae to specify the dimension from Fortran.
12: ruser($*$) – ADTYPE array User Workspace
13: ifail – Integer Input/Output

## 6Error Indicators and Warnings

c05au preserves all error codes from c05auf 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

c05au is not threaded in any implementation.

Please note that the algorithmic adjoint of Brent's method may be ill-conditioned. This means that derivatives of the zero returned in x, with respect to function parameters stored in ruser, may have limited accuracy when computed in algorithmic mode. The routine c05ay (which requires an initial interval containing the zero) can be used in symbolic mode and will compute accurate derivatives.

## 10Example

The following examples are variants of the example for c05auf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example calculates an approximation to the zero of $x-{e}^{-x}$ using a tolerance of ${\mathbf{eps}}=\text{1.0E−5}$ starting from ${\mathbf{x}}=1.0$ and using an initial search step ${\mathbf{h}}=0.1$.

Language Source File Data Results
Fortran c05au_a1w_fe.f90 None c05au_a1w_fe.r
C++ c05au_a1_algo_dcoe.cpp None c05au_a1_algo_dcoe.r
C++ c05au_a1t1_algo_dcoe.cpp None c05au_a1t1_algo_dcoe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran c05au_t1w_fe.f90 None c05au_t1w_fe.r
C++ c05au_t1_dcoe.cpp None c05au_t1_dcoe.r
C++ c05au_t2_dcoe.cpp None c05au_t2_dcoe.r

### 10.3Passive mode

Language Source File Data Results
Fortran c05au_p0w_fe.f90 None c05au_p0w_fe.r
C++ c05au_passive_dcoe.cpp None c05au_passive_dcoe.r