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## 1Purpose

d01ga is the AD Library version of the primal routine d01gaf. Based (in the C++ interface) on overload resolution, d01ga can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine d01ga_AD_f ( x, y, n, ans, er, ifail)
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail ADTYPE, Intent (In) :: x(n), y(n) ADTYPE, Intent (Out) :: ans, er 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:
C++ Interface
#include <dco.hpp>
namespace nag {
}
}
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

## 3Description

d01ga is the AD Library version of the primal routine d01gaf.
d01gaf integrates a function which is specified numerically at four or more points, over the whole of its specified range, using third-order finite difference formulae with error estimates, according to a method due to Gill and Miller (1972). For further information see Section 3 in the documentation for d01gaf.

## 4References

Gill P E and Miller G F (1972) An algorithm for the integration of unequally spaced data Comput. J. 15 80–83

## 5Arguments

In addition to the arguments present in the interface of the primal routine, d01ga 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.
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: x(n) – ADTYPE array Input
3: y(n) – ADTYPE array Input
4: n – Integer Input
5: Output
6: Output
7: ifail – Integer Input/Output

## 6Error Indicators and Warnings

d01ga preserves all error codes from d01gaf and in addition can return:
${\mathbf{ifail}}=-89$
See Error Handling in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
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.
${\mathbf{ifail}}=-444$
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

d01ga is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for d01gaf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example evaluates the integral
 $∫01 4 1+x2 dx = π$
reading in the function values at $21$ unequally spaced points.

Language Source File Data Results
Fortran d01ga_a1w_fe.f90 d01ga_a1w_fe.d d01ga_a1w_fe.r
C++ d01ga_a1w_hcppe.cpp None d01ga_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran d01ga_t1w_fe.f90 d01ga_t1w_fe.d d01ga_t1w_fe.r
C++ d01ga_t1w_hcppe.cpp None d01ga_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran d01ga_p0w_fe.f90 d01ga_p0w_fe.d d01ga_p0w_fe.r
C++ d01ga_p0w_hcppe.cpp None d01ga_p0w_hcppe.r