Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Complex (Kind=nag_wp), Intent (In)	::	z(n)
Complex (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s15drf_ (const Integer n, const Complex z[], Complex f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s15drf or nagf_specfun_erfc_complex_vector.

3 Description

s15drf computes values of the function

w (z_{i}) = e^{- {z_{i}}^{2}} erfc (- i z_{i})

, for

i = 1, 2, \dots, n

, where

erfc (z_{i})

is the complementary error function

erfc (z) = \frac{2}{\sqrt{π}} \int_{z}^{\infty} e^{- t^{2}} d t,

for complex

z

. The method used is that in Gautschi (1970) for

z

in the first quadrant of the complex plane, and is extended for

z

in other quadrants via the relations

w (- z) = 2 e^{- z^{2}} - w (z)

and

w (\bar{z}) = \bar{w (- z)}

. Following advice in Gautschi (1970) and van der Laan and Temme (1984), the code in Gautschi (1969) has been adapted to work in various precisions up to

18

decimal places. The real part of

w (z)

is sometimes known as the Voigt function.

4 References

Gautschi W (1969) Algorithm 363: Complex error function Comm. ACM 12 635

Gautschi W (1970) Efficient computation of the complex error function SIAM J. Numer. Anal. 7 187–198

van der Laan C G and Temme N M (1984) Calculation of special functions: the gamma function, the exponential integrals and error-like functions CWI Tract 10 Centre for Mathematics and Computer Science, Amsterdam

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $z (n)$ – Complex (Kind=nag_wp) array Input

On entry: the argument

z_{i}

of the function, for

i = 1, 2, \dots, n

3: $f (n)$ – Complex (Kind=nag_wp) array Output

On exit:

w (z_{i}) = e^{- {z_{i}}^{2}}

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

z_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: Real part of result overflows.
$ivalid (i) = 2$: Imaginary part of result overflows.
$ivalid (i) = 3$: Both real and imaginary part of result overflows.
$ivalid (i) = 4$: Result has less than half precision.
$ivalid (i) = 5$: Result has no precision.

5: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

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:

$ifail = 1$: On entry, at least one value of z produced a result with reduced accuracy.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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

The accuracy of the returned result depends on the argument

z_{i}

. If

z_{i}

lies in the first or second quadrant of the complex plane (i.e.,

Im (z_{i})

is greater than or equal to zero), the result should be accurate almost to machine precision, except that there is a limit of about

18

decimal places on the achievable accuracy because constants in the routine are given to this precision. With such arguments,

ivalid (i)

can only return as

ivalid (i) = 0

If however,

Im (z_{i})

is less than zero, accuracy may be lost in two ways; firstly, in the evaluation of

e^{- {z_{i}}^{2}}

, if

Im (- {z_{i}}^{2})

is large, in which case a warning will be issued through

ivalid (i) = 4

5

; and secondly, near the zeros of the required function, where precision is lost due to cancellation, in which case no warning is given – the result has absolute accuracy rather than relative accuracy. Note also that in this half-plane, one or both parts of the result may overflow – this is signalled through

ivalid (i) = 1

2

3

8 Parallelism and Performance

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

s15drf is not threaded in any implementation.

9 Further Comments

The time taken for a call of s15drf depends on the argument

z_{i}

, the time increasing as

| z_{i} | \to 0.0

s15drf may be used to compute values of

erfc (z_{i})

and

\erf z_{i}

for complex

z_{i}

by the relations

erfc (z_{i}) = e^{- {z_{i}}^{2}} w (i z_{i})

\erf z_{i} = 1 - erfc (z_{i})

. (For real arguments, s15arf and s15asf should be used.)

10 Example

This example reads values of the argument

z_{i}

from a file, evaluates the function at each value of

z_{i}

and prints the results.

s15dr: FL CL CPP AD PY MB

NAG FL Interfaces15drf (erfc_​complex_​vector)

▸▿ 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
s15drf (erfc_complex_vector)