d01rl is the AD Library version of the primal routine d01rlf. Based (in the C++ interface) on overload resolution, d01rl can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine d01rl_AD_f (

f, a, b, npts, points, epsabs, epsrel, maxsub, result, abserr, rinfo, iinfo, iuser, ruser, cpuser, ifail)

Integer, Intent (In)	::	npts, maxsub
Integer, Intent (Inout)	::	iuser(*), ifail
Integer, Intent (Out)	::	iinfo(2*max(maxsub,npts)+npts+4)
ADTYPE, Intent (In)	::	a, b, points(npts), epsabs, epsrel
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	result, abserr, rinfo(4*max(maxsub,npts)+npts+6)
Type (c_ptr), Intent (Inout)	::	ad_handle
Type (c_ptr), Intent (In)	::	cpuser
External	::	f

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 d01rl (

handle_t &ad_handle, F_T &&f, const ADTYPE &a, const ADTYPE &b, const Integer &npts, const ADTYPE points[], const ADTYPE &epsabs, const ADTYPE &epsrel, const Integer &maxsub, ADTYPE &result, ADTYPE &abserr, ADTYPE rinfo[], Integer iinfo[], 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

d01rl is the AD Library version of the primal routine d01rlf.

d01rlf is a general purpose integrator which calculates an approximation to the integral of a function

f (x)

over a finite interval

[a, b]

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

where the integrand may have local singular behaviour at a finite number of points within the integration interval. For further information see Section 3 in the documentation for d01rlf.

4 References

de Doncker E (1978) An adaptive extrapolation algorithm for automatic integration ACM SIGNUM Newsl. 13(2) 12–18

Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146

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

Wynn P (1956) On a device for computing the

e_{m} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

5 Arguments

In addition to the arguments present in the interface of the primal routine, d01rl 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.

The specification of f is:

Fortran Interface

Subroutine f (

x, nx, fv, iflag, iuser, ruser, cpuser)

Integer, Intent (In)	::	nx
Integer, Intent (Inout)	::	iflag, iuser(*)
ADTYPE, Intent (In)	::	x(nx)
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	fv(nx)
Type (c_ptr), Intent (Inout)	::	ad_handle
Type (c_ptr), Intent (In)	::	cpuser

C++ Interface

auto f = [&](

const handle_t &ad_handle, const ADTYPE x[], const Integer &nx, ADTYPE fv[], Integer &iflag)

1: ad_handle – nag::ad::handle_t Input/Output: On entry: a handle to the AD handle object.
2: x – ADTYPE array Input
3: nx – Integer Input
4: fv – ADTYPE array Output
5: iflag – Integer Input/Output
*: iuser – Integer array User Workspace
*: ruser – ADTYPE array User Workspace
*: cpuser – Type(c_ptr) User Workspace

3: a – ADTYPE Input

4: b – ADTYPE Input

5: npts – Integer Input

6: points(npts) – ADTYPE array Input

7: epsabs – ADTYPE Input

8: epsrel – ADTYPE Input

9: maxsub – Integer Input

10: result – ADTYPE Output

11: abserr – ADTYPE Output

12: rinfo( $4 \times \max (maxsub, npts) + npts + 6$ ) – ADTYPE array Output

13: iinfo( $2 \times \max (maxsub, npts) + npts + 4$ ) – Integer array Output

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

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

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

*: cpuser – Type(c_ptr) User Workspace

14: ifail – Integer Input/Output

6 Error Indicators and Warnings

d01rl preserves all error codes from d01rlf 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

d01rl is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d01rlf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example computes

\int_{0}^{1} \frac{1}{\sqrt{| x - 1 / 7 |}} d x .

A break-point is specified at

x = 1 / 7

, at which point the integrand is infinite. (For definiteness the subroutine f returns the value

0.0

at this point.)

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	d01rl_a1w_fe.f90	None	d01rl_a1w_fe.r
C++	d01rl_a1w_hcppe.cpp	None	d01rl_a1w_hcppe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	d01rl_t1w_fe.f90	None	d01rl_t1w_fe.r
C++	d01rl_t1w_hcppe.cpp	None	d01rl_t1w_hcppe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	d01rl_p0w_fe.f90	None	d01rl_p0w_fe.r
C++	d01rl_p0w_hcppe.cpp	None	d01rl_p0w_hcppe.r

NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD

NAG AD Library Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction (FL)

d01rl: FL CL CPP AD

NAG AD Libraryd01rl (dim1_fin_brkpts)

▸▿ Contents

1 Purpose