d01bd is the AD Library version of the primal routine d01bdf. Based (in the C++ interface) on overload resolution, d01bd can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine d01bd_AD_f (

ad_handle, f, a, b, epsabs, epsrel, result, abserr, iuser, ruser, ifail)

Integer, Intent (Inout)	::	iuser(*), ifail
External	::	f
ADTYPE, Intent (In)	::	a, b, epsabs, epsrel
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	result, abserr
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

C++ Interface

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

void d01bd (

handle_t &ad_handle, F_T &&f, const ADTYPE &a, const ADTYPE &b, const ADTYPE &epsabs, const ADTYPE &epsrel, ADTYPE &result, ADTYPE &abserr, 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

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

3 Description

d01bd is the AD Library version of the primal routine d01bdf.

d01bdf calculates an approximation to the integral of a function over a finite interval

[a, b]

I = \int_{a}^{b} f (x) d x .

It is non-adaptive and as such is recommended for the integration of ‘smooth’ functions. These exclude integrands with singularities, derivative singularities or high peaks on

[a, b]

, or which oscillate too strongly on

[a, b]

. For further information see Section 3 in the documentation for d01bdf.

4 References

Patterson T N L (1968) The Optimum addition of points to quadrature formulae Math. Comput. 22 847–856

Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

5 Arguments

In addition to the arguments present in the interface of the primal routine, d01bd 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: 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 $f (x)$ evaluated at x.
*: iuser( $*$ ) – Integer array User Workspace
*: ruser( $*$ ) – ADTYPE array User Workspace

3: a – ADTYPE Input

4: b – ADTYPE Input

5: epsabs – ADTYPE Input

6: epsrel – ADTYPE Input

7: result – ADTYPE Output

8: abserr – ADTYPE Output

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

User workspace.

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

User workspace.

9: ifail – Integer Input/Output

On entry: must be set to

0

- 1 or 1

On exit: any errors are indicated as described in Section 6.

6 Error Indicators and Warnings

There are no specific error codes from d01bdf, however d01bd 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

d01bd is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d01bdf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example computes

\int_{0}^{1} x^{2} \sin (10 π x) d x .

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	d01bd_a1w_fe.f90	None	d01bd_a1w_fe.r
C++	d01bd_a1w_hcppe.cpp	d01bd_a1w_hcppe.d	d01bd_a1w_hcppe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	d01bd_t1w_fe.f90	None	d01bd_t1w_fe.r
C++	d01bd_t1w_hcppe.cpp	d01bd_t1w_hcppe.d	d01bd_t1w_hcppe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	d01bd_p0w_fe.f90	None	d01bd_p0w_fe.r
C++	d01bd_p0w_hcppe.cpp	d01bd_p0w_hcppe.d	d01bd_p0w_hcppe.r

NAG Library Manual, Mark 30.3

Interfaces: FL CL CPP AD

NAG AD Library Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction (FL)

d01bd: FL CL CPP AD

NAG AD Libraryd01bd (dim1_fin_smooth)

▸▿ Contents

1 Purpose