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## 1Purpose

g01ha is the AD Library version of the primal routine g01haf. Based (in the C++ interface) on overload resolution, g01ha can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine g01ha_AD_f ( x, y, rho, p, ifail)
 Integer, Intent (Inout) :: ifail ADTYPE, Intent (In) :: x, y, rho ADTYPE, Intent (Out) :: p 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

g01ha is the AD Library 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 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.
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: Input
3: Input
4: Input
On exit: the lower tail probability for the bivariate Normal distribution.
6: ifail – Integer Input/Output

## 6Error Indicators and Warnings

g01ha preserves all error codes from g01haf 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

g01ha 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.
Description of the primal example.
This example reads values of $x$ and $y$ for a bivariate Normal distribution along with the value of $\rho$ and computes the lower tail probabilities.

Language Source File Data Results
Fortran g01ha_a1w_fe.f90 g01ha_a1w_fe.d g01ha_a1w_fe.r
C++ g01ha_a1w_hcppe.cpp g01ha_a1w_hcppe.d g01ha_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran g01ha_t1w_fe.f90 g01ha_t1w_fe.d g01ha_t1w_fe.r
C++ g01ha_t1w_hcppe.cpp g01ha_t1w_hcppe.d g01ha_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran g01ha_p0w_fe.f90 g01ha_p0w_fe.d g01ha_p0w_fe.r
C++ g01ha_p0w_hcppe.cpp g01ha_p0w_hcppe.d g01ha_p0w_hcppe.r