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## 1Purpose

e01aa is the AD Library version of the primal routine e01aaf. Based (in the C++ interface) on overload resolution, e01aa can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Subroutine e01aa_AD_f ( ad_handle, a, b, c, n1, n2, n, x, ifail)
 Integer, Intent (In) :: n1, n2, n Integer, Intent (Inout) :: ifail ADTYPE, Intent (In) :: x ADTYPE, Intent (Inout) :: a(n+1), b(n+1) ADTYPE, Intent (Out) :: c(n*(n+1)/2) 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:
#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

e01aa is the AD Library version of the primal routine e01aaf.
e01aaf interpolates a function of one variable at a given point $x$ from a table of function values ${y}_{i}$ evaluated at equidistant or non-equidistant points ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n+1$, using Aitken's technique of successive linear interpolations. For further information see Section 3 in the documentation for e01aaf.

## 4References

Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

## 5Arguments

In addition to the arguments present in the interface of the primal routine, e01aa 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.
On entry: a handle to the AD configuration data object, as created by x10aa.
2: a(${\mathbf{n}}+1$) – ADTYPE array Input/Output
3: b(${\mathbf{n}}+1$) – ADTYPE array Input/Output
4: c(${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – ADTYPE array Output
5: n1 – Integer Input
6: n2 – Integer Input
7: n – Integer Input
8: Input
9: 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 e01aaf, however e01aa can return:
${\mathbf{ifail}}=-89$
See Section 4.8.2 in the NAG AD Library Introduction for further information.
${\mathbf{ifail}}=-199$
The routine was called using a mode that has not yet been implemented.
${\mathbf{ifail}}=-443$
This check is only made if the overloaded C++ interface is used with arguments not of type double.
${\mathbf{ifail}}=-444$
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 Section 4.8.1 in the NAG AD Library Introduction for further information.

Not applicable.

## 8Parallelism and Performance

e01aa is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for e01aaf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example interpolates at $x=0.28$ the function value of a curve defined by the points
 $( xi -1.00 -0.50 0.00 0.50 1.00 1.50 yi 0.00 -0.53 -1.00 -0.46 2.00 11.09 ) .$

Language Source File Data Results
Fortran e01aa_a1w_fe.f90 e01aa_a1w_fe.d e01aa_a1w_fe.r
C++ e01aa_a1w_hcppe.cpp e01aa_a1w_hcppe.d e01aa_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran e01aa_t1w_fe.f90 e01aa_t1w_fe.d e01aa_t1w_fe.r
C++ e01aa_t1w_hcppe.cpp e01aa_t1w_hcppe.d e01aa_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran e01aa_p0w_fe.f90 e01aa_p0w_fe.d e01aa_p0w_fe.r
C++ e01aa_p0w_hcppe.cpp e01aa_p0w_hcppe.d e01aa_p0w_hcppe.r