NAG AD Library
d01fb (md_gauss)

d01fb is the AD Library version of the primal routine d01fbf. Based (in the C++ interface) on overload resolution, d01fb can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine d01fb_AD_f (

ad_handle, ndim, nptvec, lwa, weight, abscis, f, mdint, iuser, ruser, ifail)

Integer, Intent (In)	::	ndim, nptvec(ndim), lwa
Integer, Intent (Inout)	::	iuser(*), ifail
External	::	f
ADTYPE, Intent (In)	::	weight(lwa), abscis(lwa)
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	mdint
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 d01fb (

handle_t &ad_handle, const Integer &ndim, const Integer nptvec[], const Integer &lwa, const ADTYPE weight[], const ADTYPE abscis[], F_T &&f, ADTYPE &mdint, 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

d01fb is the AD Library version of the primal routine d01fbf.

d01fbf computes an estimate of a multidimensional integral (from

1

20

dimensions), given the analytic form of the integrand and suitable Gaussian weights and abscissae. For further information see Section 3 in the documentation for d01fbf.

4 References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press

5 Arguments

In addition to the arguments present in the interface of the primal routine, d01fb 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.

Note that the primal routine is a function whereas d01fb_a1w_f, is a subroutine, where the function value is returned in the additional output parameter, mdint.

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: ndim – Integer Input

3: nptvec(ndim) – Integer array Input

4: lwa – Integer Input

5: weight(lwa) – ADTYPE array Input

6: abscis(lwa) – ADTYPE array Input

7: 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, ndim, x, retval, iuser, ruser)

Integer, Intent (In)	::	ndim
Integer, Intent (Inout)	::	iuser(*)
ADTYPE, Intent (In)	::	x(ndim)
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 Integer &ndim, const ADTYPE x[], ADTYPE &retval)

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

8: mdint – ADTYPE Output

On exit: the estimate of the integral.

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

User workspace.

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

User workspace.

9: ifail – Integer Input/Output

6 Error Indicators and Warnings

d01fb preserves all error codes from d01fbf 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

d01fb is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d01fbf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example evaluates the integral

\int_{1}^{2} \int_{0}^{\infty} \int_{- \infty}^{\infty} \int_{1}^{\infty} \frac{{(x_{1} x_{2} x_{3})}^{6}}{{(x_{4} + 2)}^{8}} e^{- 2 x_{2}} e^{- 0.5 x_{3}^{2}} d x_{4} d x_{3} d x_{2} d x_{1}

using adjusted weights. The quadrature formulae chosen are:

$x_{1}$ : Gauss–Legendre, $a = 1.0$ , $b = 2.0$ ,
$x_{2}$ : Gauss–Laguerre, $a = 0.0$ , $b = 2.0$ ,
$x_{3}$ : Gauss–Hermite, $a = 0.0$ , $b = 0.5$ ,
$x_{4}$ : rational Gauss, $a = 1.0$ , $b = 2.0$ .

Four points are sufficient in each dimension, as this integral is in fact a product of four one-dimensional integrals, for each of which the chosen four-point formula is exact.