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.

2 Specification

Fortran Interface

Subroutine c05au_AD_f (

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:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w
when ADTYPE is	Type(nagad_a1t1w_w_rtype)	then AD is a1t1w
when ADTYPE is	Type(nagad_t2w_w_rtype)	then AD is t2w

C++ Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {
template <typename F_T>

void c05au (

handle_t &ad_handle, ADTYPE &x, const ADTYPE &h, const ADTYPE &eps, const ADTYPE &eta, F_T &&f, ADTYPE &a, ADTYPE &b, 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,
dco::gt1s<dco::gt1s<double>::type>::type,
dco::ga1s<dco::gt1s<double>::type>::type,

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

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.

4 References

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

5 Arguments

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.

1: ad_handle – nag::ad::handle_t Input/Output

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: x – ADTYPE Input/Output

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

3: h – ADTYPE Input

4: eps – ADTYPE Input

5: eta – ADTYPE Input

6: f – Callable Input

f needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer. If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.

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 (

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

C++ Interface

auto f = [&](

const handle_t &ad_handle, const ADTYPE &x, ADTYPE &retval)

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: x – ADTYPE Input
3: retval – ADTYPE Output: On exit: the value of $f$ at the specified point.
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

7: a – ADTYPE Output

8: b – ADTYPE Output

*: iuser( $*$ ) – Integer array User Workspace

*: ruser( $*$ ) – ADTYPE array User Workspace

Please consult Overwriting of Inputs in the NAG AD Library Introduction.

9: ifail – Integer Input/Output

6 Error Indicators and Warnings

c05au preserves all error codes from c05auf and in addition can return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.

$ifail = - 199$: 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.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

c05au is not threaded in any implementation.

9 Further Comments

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 passed in the callable f, 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.

10 Example

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

eps = 1.0E−5

starting from

x = 1.0

and using an initial search step

h = 0.1

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	c05au_a1t1w_fe.f90	None	c05au_a1t1w_fe.r
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

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	c05au_t1w_fe.f90	None	c05au_t1w_fe.r
Fortran	c05au_t2w_fe.f90	None	c05au_t2w_fe.r
C++	c05au_t1_dcoe.cpp	None	c05au_t1_dcoe.r
C++	c05au_t2_dcoe.cpp	None	c05au_t2_dcoe.r