Note: a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype. Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document. Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types. The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.

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## 1Purpose

d01ua_a1w_f is the adjoint version of the primal routine d01uaf.

## 2Specification

Fortran Interface
 Subroutine d01ua_a1w_f ( ad_handle, key, a, b, n, f, dinest, iuser, ruser, ifail)
 Integer, Intent (In) :: key, n Integer, Intent (Inout) :: iuser(*), ifail Type (nagad_a1w_w_rtype), Intent (In) :: a, b Type (nagad_a1w_w_rtype), Intent (Inout) :: ruser(*) Type (nagad_a1w_w_rtype), Intent (Out) :: dinest Type (c_ptr), Intent (Inout) :: ad_handle External :: f
The routine may be called by the names d01ua_a1w_f or nagf_quad_dim1_gauss_vec_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

d01ua_a1w_f is the adjoint 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.

## 4References

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

## 5Arguments

In addition to the arguments present in the interface of the primal routine, d01ua_a1w_f 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 – Type (c_ptr) Input/Output
On entry: a handle to the AD configuration data object, as created by x10aa_a1w_f.
2: key – Integer Input
3: Input
4: Input
5: n – Integer Input
6: f – Subroutine External Procedure
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(*) Type (nagad_a1w_w_rtype), Intent (In) :: x(nx) Type (nagad_a1w_w_rtype), Intent (Inout) :: ruser(*) Type (nagad_a1w_w_rtype), Intent (Out) :: fv(nx) Type (c_ptr), Intent (Inout) :: ad_handle
1: ad_handle – Type (c_ptr) Input/Output
On entry: a handle to the AD configuration data object.
2: Input
3: nx – Integer Input
4: Output
5: iflag – Integer Input/Output
6: iuser – Integer array User Workspace
7: User Workspace
7: Output
8: iuser($*$) – Integer array User Workspace
9: ruser($*$) – Type (nagad_a1w_w_rtype) array User Workspace
10: ifail – Integer Input/Output

## 6Error Indicators and Warnings

d01ua_a1w_f preserves all error codes from d01uaf and in addition can return:
${\mathbf{ifail}}=-89$
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-899$
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

d01ua_a1w_f is not threaded in any implementation.

None.

## 10Example

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