# NAG Library Routine Document

## 1Purpose

s01eaf evaluates the exponential function ${e}^{z}$, for complex $z$.

## 2Specification

Fortran Interface
 Function s01eaf ( z,
 Complex (Kind=nag_wp) :: s01eaf Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (In) :: z
#include nagmk26.h
 Complex s01eaf_ (const Complex *z, Integer *ifail)

## 3Description

s01eaf evaluates the exponential function ${e}^{z}$, taking care to avoid machine overflow, and giving a warning if the result cannot be computed to more than half precision. The function is evaluated as ${e}^{z}={e}^{x}\left(\mathrm{cos}y+i\mathrm{sin}y\right)$, where $x$ and $y$ are the real and imaginary parts respectively of $z$.
Since $\mathrm{cos}y$ and $\mathrm{sin}y$ are less than or equal to $1$ in magnitude, it is possible that ${e}^{x}$ may overflow although ${e}^{x}\mathrm{cos}y$ or ${e}^{x}\mathrm{sin}y$ does not. In this case the alternative formula $\mathrm{sign}\left(\mathrm{cos}y\right){e}^{x+\mathrm{ln}\left|\mathrm{cos}y\right|}$ is used for the real part of the result, and $\mathrm{sign}\left(\mathrm{sin}y\right){e}^{x+\mathrm{ln}\left|\mathrm{sin}y\right|}$ for the imaginary part. If either part of the result still overflows, a warning is returned through argument ifail.
If $\mathrm{Im}\left(z\right)$ is too large, precision may be lost in the evaluation of $\mathrm{sin}y$ and $\mathrm{cos}y$. Again, a warning is returned through ifail.

None.

## 5Arguments

1:     $\mathbf{z}$ – Complex (Kind=nag_wp)Input
On entry: the argument $z$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The real part of the result overflows, and is set to the largest safe number with the correct sign. The imaginary part of the result is meaningful.
${\mathbf{ifail}}=2$
The imaginary part of the result overflows, and is set to the largest safe number with the correct sign. The real part of the result is meaningful.
${\mathbf{ifail}}=3$
Both real and imaginary parts of the result overflow, and are set to the largest safe number with the correct signs.
${\mathbf{ifail}}=4$
The computed result is accurate to less than half precision, due to the size of $\mathrm{Im}\left(z\right)$.
${\mathbf{ifail}}=5$
The computed result has no precision, due to the size of $\mathrm{Im}\left(z\right)$, and is set to zero.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Accuracy is limited in general only by the accuracy of the standard functions in the computation of $\mathrm{sin}y$, $\mathrm{cos}y$ and ${e}^{x}$, where $x=\mathrm{Re}\left(z\right)$, $y=\mathrm{Im}\left(z\right)$. As $y$ gets larger, precision will probably be lost due to argument reduction in the evaluation of the sine and cosine functions, until the warning error ${\mathbf{ifail}}={\mathbf{4}}$ occurs when $y$ gets larger than $\sqrt{1/\epsilon }$, where $\epsilon$ is the machine precision. Note that on some machines, the intrinsic functions SIN and COS will not operate on arguments larger than about $\sqrt{1/\epsilon }$, and so ifail can never return as $4$.
In the comparatively rare event that the result is computed by the formulae $\mathrm{sign}\left(\mathrm{cos}y\right){e}^{x+\mathrm{ln}\left|\mathrm{cos}y\right|}$ and $\mathrm{sign}\left(\mathrm{sin}y\right){e}^{x+\mathrm{ln}\left|\mathrm{sin}y\right|}$, a further small loss of accuracy may be expected due to rounding errors in the logarithmic function.

## 8Parallelism and Performance

s01eaf is not threaded in any implementation.

None.

## 10Example

This example reads values of the argument $z$ from a file, evaluates the function at each value of $z$ and prints the results.

### 10.1Program Text

Program Text (s01eafe.f90)

### 10.2Program Data

Program Data (s01eafe.d)

### 10.3Program Results

Program Results (s01eafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017