# NAG FL Interfaces01eaf (exp_​complex)

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## 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 <nag.h>
 Complex s01eaf_ (const Complex *z, Integer *ifail)
The routine may be called by the names s01eaf or nagf_specfun_exp_complex.

## 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 $\mathrm{sign}\left(\mathrm{cos}y\right){e}^{x+\mathrm{ln}|\mathrm{cos}y|}$ is used for the real part of the result, and $\mathrm{sign}\left(\mathrm{sin}y\right){e}^{x+\mathrm{ln}|\mathrm{sin}y|}$ 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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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:
${\mathbf{ifail}}=1$
Argument z causes overflow in real part of result: ${\mathbf{z}}=\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
${\mathbf{ifail}}=2$
Argument z causes overflow in imaginary part of result: ${\mathbf{z}}=\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
${\mathbf{ifail}}=3$
Argument z causes overflow in both real and imaginary parts of result: ${\mathbf{z}}=\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
${\mathbf{ifail}}=4$
The imaginary part of argument z is so large that the result is accurate to less than half precision: ${\mathbf{z}}=\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
${\mathbf{ifail}}=5$
The imaginary part of argument z is so large that the result has no precision: ${\mathbf{z}}=\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 $\mathrm{sign}\left(\mathrm{cos}y\right){e}^{x+\mathrm{ln}|\mathrm{cos}y|}$ and $\mathrm{sign}\left(\mathrm{sin}y\right){e}^{x+\mathrm{ln}|\mathrm{sin}y|}$, 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)