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

d01bd_a1w_f is the adjoint version of the primal routine d01bdf.

## 2Specification

Fortran Interface
 Subroutine d01bd_a1w_f ( ad_handle, f, a, b, epsabs, epsrel, result, abserr, iuser, ruser, ifail)
 Integer, Intent (Inout) :: iuser(*), ifail External :: f Type (nagad_a1w_w_rtype), Intent (In) :: a, b, epsabs, epsrel Type (nagad_a1w_w_rtype), Intent (Inout) :: ruser(*) Type (nagad_a1w_w_rtype), Intent (Out) :: result, abserr Type (c_ptr), Intent (Inout) :: ad_handle
The routine may be called by the names d01bd_a1w_f or nagf_quad_dim1_fin_smooth_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

d01bd_a1w_f is the adjoint version of the primal routine d01bdf.
d01bdf calculates an approximation to the integral of a function over a finite interval $\left[a,b\right]$:
 $I= ∫ab f(x) dx .$
It is non-adaptive and as such is recommended for the integration of ‘smooth’ functions. These exclude integrands with singularities, derivative singularities or high peaks on $\left[a,b\right]$, or which oscillate too strongly on $\left[a,b\right]$. For further information see Section 3 in the documentation for d01bdf.
Patterson T N L (1968) The Optimum addition of points to quadrature formulae Math. Comput. 22 847–856
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

## 5Arguments

In addition to the arguments present in the interface of the primal routine, d01bd_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: f – Subroutine External Procedure
Note that f is a subroutine in this interface, returning the function value via the additional output parameter retval.
The specification of f is:
Fortran Interface
 Subroutine f ( ad_handle, x, retval, iuser, ruser)
 Integer, Intent (Inout) :: iuser(*) Type (nagad_a1w_w_rtype), Intent (In) :: x Type (nagad_a1w_w_rtype), Intent (Inout) :: ruser(*) Type (nagad_a1w_w_rtype), Intent (Out) :: retval 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
On exit: the value of $f\left(x\right)$ evaluated at x.
4: iuser($*$) – Integer array User Workspace
5: ruser($*$)Type (nagad_a1w_w_rtype) array User Workspace
3: Input
4: Input
5: Input
6: Input
7: Output
8: Output
9: iuser($*$) – Integer array User Workspace
User workspace.
10: ruser($*$) – Type (nagad_a1w_w_rtype) array User Workspace
User workspace.
11: ifail – Integer Input/Output
On entry: must be set to $\mathrm{0}$, $-\mathrm{1}\text{ or }\mathrm{1}$. On exit: any errors are indicated as described in Section 6.

## 6Error Indicators and Warnings

There are no specific error codes from d01bdf, however d01bd_a1w_f 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

d01bd_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for d01bdf, modified to demonstrate calling the NAG AD Library.