NAG AD Library
d01ua (dim1_gauss_vec)

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1 Purpose

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

Fortran Interface
Subroutine d01ua_AD_f ( ad_handle, key, a, b, n, f, dinest, iuser, ruser, ifail)
Integer, Intent (In) :: key, n
Integer, Intent (Inout) :: iuser(*), ifail
ADTYPE, Intent (In) :: a, b
ADTYPE, Intent (Inout) :: ruser(*)
ADTYPE, Intent (Out) :: dinest
Type (c_ptr), Intent (Inout) :: ad_handle
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 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:
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_handlenag::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: aADTYPE Input
4: bADTYPE 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:
Fortran Interface
Subroutine f ( ad_handle, x, nx, fv, iflag, iuser, ruser)
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
C++ Interface
auto f = [&]( const handle_t &ad_handle, const ADTYPE x[], const Integer &nx, ADTYPE fv[], Integer &iflag)
1: ad_handlenag::ad::handle_t Input/Output
On entry: a handle to the AD handle object.
2: xADTYPE array Input
3: nx – Integer Input
4: fvADTYPE array Output
5: iflag – Integer Input/Output
*: iuser – Integer array User Workspace
*: ruserADTYPE array User Workspace
7: dinestADTYPE 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:
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.
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.
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
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


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
0141+x2 dx=π  
by Gauss–Legendre quadrature;
2 1x2 lnx dx =0.378671  
by rational Gauss quadrature with b=0;
by Gauss–Laguerre quadrature with b=1; and
- +e-3x2-4x-1dx=- +e-3 (x+1) 2e2x+2dx=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