NAG CL Interface
s15ddc (erfc_​complex)

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1 Purpose

s15ddc computes values of the function w(z)=e-z2erfc(-iz), for Complex z.

2 Specification

#include <nag.h>
Complex  s15ddc (Complex z, NagError *fail)
The function may be called by the names: s15ddc, nag_specfun_erfc_complex or nag_complex_erfc.

3 Description

s15ddc computes values of the function w(z)=e-z2erfc(-iz), where erfc(z) is the complementary error function
erfc(z) = 2π z e-t2 dt ,  
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)=2e-z2-w(z) and w(z¯)=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: z Complex Input
On entry: the argument z of the function.
2: 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_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.
NE_RESULT_HALF_PRECISION
Result has less than half precision when entered with argument z=(value,value).
NE_RESULT_IMAGINARY_OVERFLOW
Imaginary part of result overflows when entered with argument z=(value,value).
NE_RESULT_NO_PRECISION
Result has no precision when entered with argument z=(value,value).
NE_RESULT_OVERFLOW
Both real and imaginary parts of result overflow when entered with argument z=(value,value).
NE_RESULT_REAL_OVERFLOW
Real part of result overflows when entered with argument z=(value,value).

7 Accuracy

The accuracy of the returned result depends on the argument z. If z lies in the first or second quadrant of the complex plane (i.e., Im(z) 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, fail can only return as fail.code= NE_NOERROR.
If however, Im(z) is less than zero, accuracy may be lost in two ways; firstly, in the evaluation of e-z2, if Im(-z2) is large, in which case a warning will be issued through fail.code= NE_RESULT_HALF_PRECISION or NE_RESULT_NO_PRECISION; 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 fail.code= NE_RESULT_IMAGINARY_OVERFLOW, NE_RESULT_OVERFLOW or NE_RESULT_REAL_OVERFLOW.

8 Parallelism and Performance

s15ddc is not threaded in any implementation.

9 Further Comments

The time taken for a call of s15ddc depends on the argument z, the time increasing as |z|0.0.
s15ddc may be used to compute values of erfc(z) and erfz for Complex z by the relations erfc(z)=e-z2w(iz), erfz=1-erfc(z). (For double arguments, s15adc and s15aec should be used.)

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.

10.1 Program Text

Program Text (s15ddce.c)

10.2 Program Data

Program Data (s15ddce.d)

10.3 Program Results

Program Results (s15ddce.r)