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

g01ha_a1w_f is the adjoint version of the primal routine g01haf.

## 2Specification

Fortran Interface
 Subroutine g01ha_a1w_f ( ad_handle, x, y, rho, p, ifail)
 Integer, Intent (Inout) :: ifail Type (nagad_a1w_w_rtype), Intent (In) :: x, y, rho Type (nagad_a1w_w_rtype), Intent (Out) :: p Type (c_ptr), Intent (Inout) :: ad_handle
The routine may be called by the names g01ha_a1w_f or nagf_stat_prob_bivariate_normal_a1w. The corresponding t1w and p0w variants of this routine are also available.

## 3Description

g01ha_a1w_f is the adjoint version of the primal routine g01haf.
g01haf returns the lower tail probability for the bivariate Normal distribution. For further information see Section 3 in the documentation for g01haf.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities Statistics and Computing 14 151–160
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

## 5Arguments

In addition to the arguments present in the interface of the primal routine, g01ha_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.
Note that the primal routine is a function whereas g01ha_a1w_f, is a subroutine, where the function value is returned in the additional output parameter, p.
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: Input
3: Input
4: Input
5: Output
On exit: the lower tail probability for the bivariate Normal distribution.
6: ifail – Integer Input/Output

## 6Error Indicators and Warnings

g01ha_a1w_f preserves all error codes from g01haf 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

g01ha_a1w_f is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for g01haf, modified to demonstrate calling the NAG AD Library.