NAG Library Routine Document
s01eaf (exp_complex)
1
Purpose
s01eaf evaluates the exponential function ${e}^{z}$, for
complex
$z$.
2
Specification
Fortran Interface
Complex (Kind=nag_wp)  ::  s01eaf  Integer, Intent (Inout)  ::  ifail  Complex (Kind=nag_wp), Intent (In)  ::  z 

C Header Interface
#include nagmk26.h
Complex 
s01eaf_ (const Complex *z, Integer *ifail) 

3
Description
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.
4
References
None.
5
Arguments
 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).
6
Error 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.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
7
Accuracy
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.
8
Parallelism and Performance
s01eaf is not threaded in any implementation.
None.
10
Example
This example reads values of the argument $z$ from a file, evaluates the function at each value of $z$ and prints the results.
10.1
Program Text
Program Text (s01eafe.f90)
10.2
Program Data
Program Data (s01eafe.d)
10.3
Program Results
Program Results (s01eafe.r)