NAG FL Interface
s15ddf (erfc_​complex)

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1 Purpose

s15ddf computes values of the function w(z)=e-z2erfc(-iz), for complex z.

2 Specification

Fortran Interface
Function s15ddf ( z, ifail)
Complex (Kind=nag_wp) :: s15ddf
Integer, Intent (Inout) :: ifail
Complex (Kind=nag_wp), Intent (In) :: z
C Header Interface
#include <nag.h>
Complex  s15ddf_ (const Complex *z, Integer *ifail)
The routine may be called by the names s15ddf or nagf_specfun_erfc_complex.

3 Description

s15ddf 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 (Kind=nag_wp) Input
On entry: the argument z of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 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 or 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Real part of result overflows when entered with argument z=(value,value).
Imaginary part of result overflows when entered with argument z=(value,value).
Both real and imaginary parts of result overflow when entered with argument z=(value,value).
Result has less than half precision when entered with argument z=(value,value).
Result has no precision when entered with argument z=(value,value).
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.
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.
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. 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 routine are given to this precision. With such arguments, ifail can only return as ifail=0.
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 ifail=4 or 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 ifail=1, 2 or 3.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s15ddf is not threaded in any implementation.

9 Further Comments

The time taken for a call of s15ddf depends on the argument z, the time increasing as |z|0.0.
s15ddf 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 real arguments, s15adf and s15aef 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 (s15ddfe.f90)

10.2 Program Data

Program Data (s15ddfe.d)

10.3 Program Results

Program Results (s15ddfe.r)