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 (

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:

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++ Header Interface

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

void c05au (

void *&ad_handle, ADTYPE &x, const ADTYPE &h, const ADTYPE &eps, const ADTYPE &eta,
void (NAG_CALL f)(void *&ad_handle, const ADTYPE &x, ADTYPE &retval, Integer iuser[], ADTYPE ruser[]),
ADTYPE &a, ADTYPE &b, const Integer &liuser, Integer iuser[], const Integer &lruser, ADTYPE ruser[], 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 – Pointer to AD Data Input/Output

On entry: a handle to the AD configuration data object, as created by x10aa.

2: x – ADTYPE Input/Output

3: h – ADTYPE Input

4: eps – ADTYPE Input

5: eta – ADTYPE 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

C++ Header Interface

void f (

void *&ad_handle, const ADTYPE &x, ADTYPE &retval, Integer iuser[], ADTYPE ruser[])

1: ad_handle – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object.
2: x – ADTYPE Input
3: retval – ADTYPE Output: On exit: the value of $f$ at the specified point.
4: iuser – Integer array User Workspace
5: ruser – ADTYPE array User Workspace

7: a – ADTYPE Output

8: b – ADTYPE 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

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 Section 4.8.2 in the NAG AD Library Introduction for further information.

$ifail = - 199$: The routine was called using a mode that has not yet been implemented.

$ifail = - 443$: On entry: ad_handle is nullptr.
This check is only made if the overloaded C++ interface is used with arguments not of type double.

$ifail = - 444$: 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 Section 4.8.1 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 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.

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_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
C++	c05au_t1_dcoe.cpp	None	c05au_t1_dcoe.r
C++	c05au_t2_dcoe.cpp	None	c05au_t2_dcoe.r

10.3 Passive 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

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG AD Library Introduction

C05 (Roots) Chapter Contents

C05 (Roots) Chapter Introduction (FL)

c05au: FL CL CPP AD

NAG AD Libraryc05au (contfn_brent_interval)

▸▿ Contents

1 Purpose