NAG AD Library
c05ay (contfn_brent)

c05ay is the AD Library version of the primal routine c05ayf. Based (in the C++ interface) on overload resolution, c05ay can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first and second order. The parameter ad_handle can be used to choose whether adjoints are computed using a symbolic adjoint or straightforward algorithmic differentiation. In addition, the routine has further optimisations when expert symbolic mode is selected.

2 Specification

Fortran Interface

Subroutine c05ay_AD_f (

ad_handle, a, b, eps, eta, f, x, iuser, ruser, ifail)

Integer, Intent (Inout)	::	iuser(*), ifail
External	::	f
ADTYPE, Intent (In)	::	a, b, eps, eta
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	x
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 c05ay (

void *&ad_handle, const ADTYPE &a, const ADTYPE &b, const ADTYPE &eps, const ADTYPE &eta,
void (NAG_CALL f)(void *&ad_handle, const ADTYPE &x, ADTYPE &retval, Integer iuser[], ADTYPE ruser[]),
ADTYPE &x, 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

c05ay is the AD Library version of the primal routine c05ayf.

c05ayf locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. For further information see Section 3 in the documentation for c05ayf.

3.1 Symbolic Adjoints

c05ay can provide symbolic adjoints.

The symbolic adjoints assumes

(i)successful computation of primal problem ( $ifail = 0$ or $2$ on exit of c05ayf), i.e.,

$f (\hat{x}, p) = 0,$ (1)

where $\hat{x}$ is a solution;
(ii)the first derivative at the solution is not equal zero

$\nabla_{x} f (\hat{x}, p) \neq 0 .$ (2)

Here

p

is a placeholder for any user variable either passed via ruser or via global variables.

3.1.1 Symbolic Mode

Symbolic mode may be selected by calling x10ac with mode set to nagad_symbolic prior to calling c05ay. No further changes are needed compared to using algorithmic mode.

3.1.2 Symbolic Expert Mode

Symbolic expert mode may be selected by calling x10ac with mode set to nagad_symbolic_expert prior to calling c05ay. In comparison to the symbolic mode, in nagad_symbolic_expert mode the user-supplied primal callback needs a specific implementation to support symbolic computation, but this can improve overall performance. See the example c05ay_a1_sym_expert_dcoe.cpp for details.

3.1.3 Mathematical Background

The symbolic adjoint computes

z = - \frac{x_{(1)}}{\nabla_{x} f (\hat{x}, p)}

followed by an adjoint projection through the user-supplied adjoint routine

p_{(1)} = p_{(1)} + \nabla_{p} f (\hat{x}, p) \cdot z .

Both

\nabla_{x} f (x, p)

as well as

\nabla_{p} f (x, p)

are computed using the user-supplied adjoint routine.

Please see Du Toit and Naumann (2017), Naumann et al. (2017) and Giles (2017) for reference.

3.1.4 Usable Adjoints

You can set or access the adjoints of output argument x. The adjoints of all other output arguments are ignored.

c05ay increments the adjoints of the variable

p

, where

p

is given by the argument ruser or by use of globals (see (3)).

The adjoints of all other input parameters are not referenced.

4 References

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

Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance

Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation

Naumann U, Lotz J, Leppkes K and Towara M (2017) Algorithmic Differentiation of Numerical Methods: Tangent and Adjoint Solvers for Parameterized Systems of Nonlinear Equations

5 Arguments

In addition to the arguments present in the interface of the primal routine, c05ay 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. The computational mode may be selected by calling x10ac with this handle.

2: a – ADTYPE Input

3: b – 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 $f$ evaluated at x.
4: iuser – Integer array User Workspace
5: ruser – ADTYPE array User Workspace

7: x – ADTYPE Output

8: liuser Input

User workspace dimension (C++ only), see x10af to specify the dimension from Fortran.

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

10: lruser Input

User workspace dimension (C++ only), see x10ae to specify the dimension from Fortran.

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

12: ifail – Integer Input/Output

6 Error Indicators and Warnings

c05ay preserves all error codes from c05ayf 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

c05ay 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. This routine can be used in symbolic mode which will compute accurate derivatives.

10 Example

The following examples are variants of the example for c05ayf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example calculates an approximation to the zero of

e^{- x} - x

within the interval

[0, 1]

using a tolerance of

eps = 1.0E−5

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	c05ay_a1w_fe.f90	c05ay_a1w_fe.d	c05ay_a1w_fe.r
C++	c05ay_a1_algo_dcoe.cpp	None	c05ay_a1_algo_dcoe.r
C++	c05ay_a1_sym_dcoe.cpp	None	c05ay_a1_sym_dcoe.r
C++	c05ay_a1_sym_expert_dcoe.cpp	None	c05ay_a1_sym_expert_dcoe.r
C++	c05ay_a1t1_algo_dcoe.cpp	None	c05ay_a1t1_algo_dcoe.r
C++	c05ay_a1t1_sym_dcoe.cpp	None	c05ay_a1t1_sym_dcoe.r
C++	c05ay_a1t1_sym_expert_dcoe.cpp	None	c05ay_a1t1_sym_expert_dcoe.r