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

d01tb_a1w_f is the adjoint version of the primal routine d01tbf.

## 2Specification

Fortran Interface
 Subroutine d01tb_a1w_f ( ad_handle, key, a, b, n, weight, abscis, ifail)
 Integer, Intent (In) :: key, n Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (In) :: a, b Type (nagad_a1w_w_rtype), Intent (Out) :: weight(n), abscis(n) Type (c_ptr), Intent (Inout) :: ad_handle
The routine may be called by the names d01tb_a1w_f or nagf_quad_dim1_gauss_wres_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

d01tb_a1w_f is the adjoint version of the primal routine d01tbf.
d01tbf returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite. For further information see Section 3 in the documentation for d01tbf.
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, d01tb_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: weight(n) – Type (nagad_a1w_w_rtype) array Output
7: abscis(n) – Type (nagad_a1w_w_rtype) array Output
8: ifail – Integer Input/Output

## 6Error Indicators and Warnings

d01tb_a1w_f preserves all error codes from d01tbf 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

d01tb_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for d01tbf, modified to demonstrate calling the NAG AD Library.