, 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.

References

None.

Parameters

Compulsory Input Parameters

1: $z$ – complex scalar: The argument $z$ of the function.

Optional Input Parameters

None.

Output Parameters

1: $result$ – complex scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: 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.

W $ifail = 2$: 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.

W $ifail = 3$: Both real and imaginary parts of the result overflow, and are set to the largest safe number with the correct signs.

W $ifail = 4$: The computed result is accurate to less than half precision, due to the size of $Im (z)$ .

$ifail = 5$: The computed result has no precision, due to the size of $Im (z)$ , and is set to zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

Further Comments

None.

Example

This example reads values of the argument

z

from a file, evaluates the function at each value of

z

and prints the results.

Open in the MATLAB editor: s01ea_example

function s01ea_example


fprintf('s01ea example results\n\n');

n = 4;
z(1:n) = [1 + 1.e-9i    -0.5 + 2i     0.0 - 2i    -2.5 - 1.5i];
result = z;

for j=1:n
  [result(j), ifail] = s01ea(z(j));
end

disp('         z                 exp(z)');
disp([z; result]');

s01ea example results

         z                 exp(z)
   1.0000 - 0.0000i   2.7183 - 0.0000i
  -0.5000 - 2.0000i  -0.2524 - 0.5515i
   0.0000 + 2.0000i  -0.4161 + 0.9093i
  -2.5000 + 1.5000i   0.0058 + 0.0819i

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox