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## 1Purpose

f01fc is the AD Library version of the primal routine f01fcf. Based (in the C++ interface) on overload resolution, f01fc can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

## 2Specification

Fortran Interface
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail ADCTYPE, Intent (Inout) :: a(lda,*) 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 f01fc ( void *&ad_handle, const Integer &n, ADCTYPE a[], const Integer &lda, 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

f01fc is the AD Library version of the primal routine f01fcf.
f01fcf computes the matrix exponential, ${e}^{A}$, of a complex $n×n$ matrix $A$. For further information see Section 3 in the documentation for f01fcf.

## 4References

Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev. 45 3–49

## 5Arguments

In addition to the arguments present in the interface of the primal routine, f01fc 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: n – Integer Input
3: a(lda, $*$) – ADCTYPE array Input/Output
4: lda – Integer Input
5: ifail – Integer Input/Output

## 6Error Indicators and Warnings

f01fc preserves all error codes from f01fcf 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

f01fc is not threaded in any implementation.

None.

## 10Example

The following examples are variants of the example for f01fcf, modified to demonstrate calling the NAG AD Library.
Description of the primal example.
This example finds the matrix exponential of the matrix
 $A = ( 1+2i 2+2i 2+2i 2+i 3+2i 1 1 2+i 3+2i 2+2i 1 2+i 3+2i 3+2i 3+2i 1+i ) .$

Language Source File Data Results
Fortran f01fc_a1w_fe.f90 f01fc_a1w_fe.d f01fc_a1w_fe.r
C++ f01fc_a1w_hcppe.cpp f01fc_a1w_hcppe.d f01fc_a1w_hcppe.r

### 10.2Tangent modes

Language Source File Data Results
Fortran f01fc_t1w_fe.f90 f01fc_t1w_fe.d f01fc_t1w_fe.r
C++ f01fc_t1w_hcppe.cpp f01fc_t1w_hcppe.d f01fc_t1w_hcppe.r

### 10.3Passive mode

Language Source File Data Results
Fortran f01fc_p0w_fe.f90 f01fc_p0w_fe.d f01fc_p0w_fe.r
C++ f01fc_p0w_hcppe.cpp f01fc_p0w_hcppe.d f01fc_p0w_hcppe.r