d01ua is the AD Library version of the primal routine d01uaf. Based (in the C++ interface) on overload resolution, d01ua can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

C++ Interface

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

void d01ua (

handle_t &ad_handle, const Integer &key, const ADTYPE &a, const ADTYPE &b, const Integer &n, F_T &&f, ADTYPE &dinest, 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

d01ua is the AD Library version of the primal routine d01uaf.

d01uaf computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss–Legendre), a semi-infinite interval (Gauss–Laguerre, rational Gauss), and an infinite interval (Gauss–Hermite). For further information see Section 3 in the documentation for d01uaf.

4 References

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

Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

Ralston A (1965) A First Course in Numerical Analysis pp. 87–90 McGraw–Hill

Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall

5 Arguments

In addition to the arguments present in the interface of the primal routine, d01ua 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: key – Integer Input

3: a – ADTYPE Input

4: b – ADTYPE Input

5: n – Integer 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.

The specification of f is:

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

7: dinest – 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

d01ua preserves all error codes from d01uaf 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

d01ua is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d01uaf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

This example evaluates the integrals

\int_{0}^{1} \frac{4}{1 + x^{2}} d x = π

by Gauss–Legendre quadrature;

\int_{2}^{\infty} \frac{1}{x^{2} \ln x} d x = 0.378671

by rational Gauss quadrature with

b = 0

;

\int_{2}^{\infty} \frac{e^{- x}}{x} d x = 0.048901

by Gauss–Laguerre quadrature with

b = 1

; and

\int_{- \infty}^{+ \infty} e^{- 3 x^{2} - 4 x - 1} d x = \int_{- \infty}^{+ \infty} e^{- 3 {(x + 1)}^{2}} e^{2 x + 2} d x = 1.428167

by Gauss–Hermite quadrature with

a = −1

and

b = 3

The formulae with

n = 2, 4, 8, 16, 32

and

64

are used in each case. Both adjusted and normal weights are used for Gauss–Laguerre and Gauss–Hermite quadrature.

10.1 Adjoint modes

Language	Source File	Data	Results
Fortran	d01ua_a1w_fe.f90	None	d01ua_a1w_fe.r
C++	d01ua_a1w_hcppe.cpp	None	d01ua_a1w_hcppe.r

10.2 Tangent modes

Language	Source File	Data	Results
Fortran	d01ua_t1w_fe.f90	None	d01ua_t1w_fe.r
C++	d01ua_t1w_hcppe.cpp	None	d01ua_t1w_hcppe.r

10.3 Passive mode

Language	Source File	Data	Results
Fortran	d01ua_p0w_fe.f90	None	d01ua_p0w_fe.r
C++	d01ua_p0w_hcppe.cpp	None	d01ua_p0w_hcppe.r

NAG Library Manual, Mark 27.3

Interfaces: FL CL CPP AD

NAG AD Library Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction (FL)

d01ua: FL CL CPP AD

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

NAG AD Libraryd01ua (dim1_gauss_vec)

▸▿ Contents

1 Purpose