nag_complex_erfc (s15ddc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_complex_erfc (s15ddc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nags.h>
Complex  nag_complex_erfc (Complex z, NagError *fail)

3  Description

nag_complex_erfc (s15ddc) computes values of the function wz=e-z2erfc-iz, where erfcz is the complementary error function
erfcz=2πze-t2dt,
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-wz and wz¯=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 wz 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:     zComplexInput
On entry: the argument z of the function.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
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., Imz 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 Imz 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  Further Comments

The time taken for a call of nag_complex_erfc (s15ddc) depends on the argument z, the time increasing as z0.0.
nag_complex_erfc (s15ddc) may be used to compute values of erfcz and erfz for Complex z by the relations erfcz=e-z2wiz, erfz=1-erfcz. (For double arguments, nag_erfc (s15adc) and nag_erf (s15aec) should be used.)

9  Example

This example reads values of the argument z from a file, evaluates the function at each value of z and prints the results.

9.1  Program Text

Program Text (s15ddce.c)

9.2  Program Data

Program Data (s15ddce.d)

9.3  Program Results

Program Results (s15ddce.r)


nag_complex_erfc (s15ddc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012