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## 1Purpose

s18de is the AD Library version of the primal routine s18def. Based (in the C++ interface) on overload resolution, s18de can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: nz ADTYPE, Intent (In) :: fnu ADCTYPE, Intent (In) :: z ADCTYPE, Intent (Out) :: cy(n) Character (1), Intent (In) :: scal 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 {
 void s18de ( void *&ad_handle, const ADTYPE &fnu, const ADCTYPE &z, const Integer &n, const char *scal, ADCTYPE cy[], Integer &nz, Integer &ifail)
}
}
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

s18de is the AD Library version of the primal routine s18def.
s18def returns a sequence of values for the modified Bessel functions ${I}_{\nu +n}\left(z\right)$ for complex $z$, non-negative $\nu$ and $n=0,1,\dots ,N-1$, with an option for exponential scaling. For further information see Section 3 in the documentation for s18def.

## 4References

NIST Digital Library of Mathematical Functions
Amos D E (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order ACM Trans. Math. Software 12 265–273

## 5Arguments

In addition to the arguments present in the interface of the primal routine, s18de 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: Input
4: n – Integer Input
5: scal – character Input
6: cy(n) – ADCTYPE array Output
7: nz – Integer Output
8: ifail – Integer Input/Output

## 6Error Indicators and Warnings

s18de preserves all error codes from s18def and in addition 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

s18de is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for s18def, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example prints a caption and then proceeds to read sets of data from the input data stream. The first datum is a value for the order fnu, the second is a complex value for the argument, z, and the third is a character value to set the argument scal. The program calls the routine with ${\mathbf{n}}=2$ to evaluate the function for orders fnu and ${\mathbf{fnu}}+1$, and it prints the results. The process is repeated until the end of the input data stream is encountered.

Language Source File Data Results
Fortran s18de_a1w_fe.f90 None s18de_a1w_fe.r
C++ s18de_a1w_hcppe.cpp None s18de_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran s18de_t1w_fe.f90 None s18de_t1w_fe.r
C++ s18de_t1w_hcppe.cpp None s18de_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran s18de_p0w_fe.f90 None s18de_p0w_fe.r
C++ s18de_p0w_hcppe.cpp None s18de_p0w_hcppe.r