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 symbolic expert strategy 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++ Interface

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

void c05ay (

handle_t &ad_handle, const ADTYPE &a, const ADTYPE &b, const ADTYPE &eps, const ADTYPE &eta, F_T &&f, ADTYPE &x, 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,
dco::gt1v<double, N>::type,
dco::gt1s<dco::gt1v<double, N>::type>::type,
dco::ga1v<double, N>::type,
dco::ga1v<dco::gt1v<double, N>::type, M>::type,
dco::ga1s<dco::gt1v<double, N>::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 for all first and second order scalar modes. Vector modes are currently not supported.

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 passed in the callable f.

3.1.1 Symbolic Strategy

Symbolic strategy may be selected by calling ad_handle.set_strategy(nag::ad::symbolic) prior to calling c05ay. No further changes are needed compared to using the algorithmic strategy.

3.1.2 Symbolic Expert Strategy

Symbolic expert strategy may be selected by calling ad_handle.set_strategy(nag::ad::symbolic_expert) prior to calling c05ay. In contrast to the symbolic strategy, in symbolic expert strategy 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 passed in the callable f (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 – 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 and AD Strategies in the NAG AD Library Introduction.

2: a – ADTYPE Input

3: b – 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 (

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++ 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 $f$ evaluated at x.
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace

7: x – ADTYPE Output

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

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

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

8: 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 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

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 passed in the callable f, may have limited accuracy when computed in algorithmic mode. This routine can be used in symbolic mode which will compute accurate derivatives.

In addition to the dco/c++ scalar types, this routine supports a subset of the vector types. Those types are only supported for the C++ interface. The underlying implementation currently falls back onto the scalar type. This means there is no performance enhancement to be expected from using the vector types. At the moment, these interfaces are provided to enable a seamless integration of vector types used to increase performance elsewhere in your code.

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 scalar modes

Language	Source File	Data	Results
Fortran	c05ay_a1t1w_fe.f90	c05ay_a1t1w_fe.d	c05ay_a1t1w_fe.r
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

10.2 Tangent scalar modes

Language	Source File	Data	Results
Fortran	c05ay_t1w_fe.f90	c05ay_t1w_fe.d	c05ay_t1w_fe.r
Fortran	c05ay_t2w_fe.f90	c05ay_t2w_fe.d	c05ay_t2w_fe.r
C++	c05ay_t1_dcoe.cpp	None	c05ay_t1_dcoe.r
C++	c05ay_t2_dcoe.cpp	None	c05ay_t2_dcoe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	c05ay_p0w_fe.f90	c05ay_p0w_fe.d	c05ay_p0w_fe.r
C++	c05ay_passive_dcoe.cpp	None	c05ay_passive_dcoe.r

10.4 Vector modes

Language	Source File	Data	Results
C++	c05ay_a1v_dcoe.cpp	None	c05ay_a1v_dcoe.r
C++	c05ay_t1v_dcoe.cpp	None	c05ay_t1v_dcoe.r
C++	c05ay_t2va1_dcoe.cpp	None	c05ay_t2va1_dcoe.r
C++	c05ay_t2va1v_dcoe.cpp	None	c05ay_t2va1v_dcoe.r
C++	c05ay_t2vt1_dcoe.cpp	None	c05ay_t2vt1_dcoe.r