. 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: $z [n]$ – const Complex Input

On entry: the argument

z_{i}

of the function, for

i = 1, 2, \dots, n

3: $f [n]$ – Complex Output

On exit:

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

, the function values.

4: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for

z_{i}

, for

i = 1, 2, \dots, n

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

5: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of z produced a result with reduced accuracy.
Check ivalid for more 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 function are given to this precision. With such arguments,

ivalid [i - 1]

can only return as

ivalid [i - 1] = 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 - 1] = 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] = 1

2

3

8 Parallelism and Performance

s15drc is not threaded in any implementation.

9 Further Comments

The time taken for a call of s15drc depends on the argument

z_{i}

, the time increasing as

|z_{i}| \to 0.0

s15drc 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 double arguments, s15arc and s15asc 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.

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15dr: FL CL AD

NAG CL Interfaces15drc (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 CL Interface
s15drc (erfc_complex_vector)