, 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} (\cos y + i \sin y)

, where

x

and

y

are the real and imaginary parts respectively of

z

Since

\cos y

and

\sin y

are less than or equal to

1

in magnitude, it is possible that

e^{x}

may overflow although

e^{x} \cos y

e^{x} \sin y

does not. In this case the alternative formula

sign (\cos y) e^{x + \ln | \cos y |}

is used for the real part of the result, and

sign (\sin y) e^{x + \ln | \sin y |}

for the imaginary part. If either part of the result still overflows, a warning is returned through argument ifail.

Im (z)

is too large, precision may be lost in the evaluation of

\sin y

and

\cos y

. Again, a warning is returned through ifail.

4 References

None.

5 Arguments

1: $z$ – Complex (Kind=nag_wp) Input: On entry: the argument $z$ of the function.
2: $ifail$ – 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 $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: Argument z causes overflow in real part of result: $z = (⟨ value ⟩, ⟨ value ⟩)$ .

$ifail = 2$: Argument z causes overflow in imaginary part of result: $z = (⟨ value ⟩, ⟨ value ⟩)$ .

$ifail = 3$: Argument z causes overflow in both real and imaginary parts of result: $z = (⟨ value ⟩, ⟨ value ⟩)$ .

$ifail = 4$: The imaginary part of argument z is so large that the result is accurate to less than half precision: $z = (⟨ value ⟩, ⟨ value ⟩)$ .

$ifail = 5$: The imaginary part of argument z is so large that the result has no precision: $z = (⟨ value ⟩, ⟨ value ⟩)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Accuracy is limited in general only by the accuracy of the standard functions in the computation of

\sin y

\cos y

and

e^{x}

, where

x = Re (z)

y = Im (z)

. 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

ifail = 4

occurs when

y

gets larger than

\sqrt{1 / ε}

, where

ε

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 / ε}

, and so ifail can never return as

4

In the comparatively rare event that the result is computed by the formulae

sign (\cos y) e^{x + \ln | \cos y |}

and

sign (\sin y) e^{x + \ln | \sin y |}

, a further small loss of accuracy may be expected due to rounding errors in the logarithmic function.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

s01eaf is not threaded in any implementation.

9 Further Comments

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.

s01ea: FL CL CPP AD PY MB

NAG FL Interfaces01eaf (exp_​complex)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s01eaf (exp_complex)