NAG FL Interface
s15drf (erfc_​complex_​vector)

1 Purpose

s15drf computes values of the function wz=e-z2erfc-iz, for an array of complex values z.

2 Specification

Fortran Interface
Subroutine s15drf ( n, z, f, ivalid, ifail)
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 wzi=e-zi2erfc-izi, for i=1,2,,n, where erfczi is the complementary error function
erfcz = 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-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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: zn Complex (Kind=nag_wp) array Input
On entry: the argument zi of the function, for i=1,2,,n.
3: fn Complex (Kind=nag_wp) array Output
On exit: wzi=e-zi2, the function values.
4: ivalidn Integer array Output
On exit: ivalidi contains the error code for zi, for i=1,2,,n.
No error.
Real part of result overflows.
Imaginary part of result overflows.
Both real and imaginary part of result overflows.
Result has less than half precision.
Result has no precision.
5: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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:
On entry, at least one value of z produced a result with reduced accuracy.
Check ivalid for more information.
On entry, n=value.
Constraint: n0.
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 zi. If zi lies in the first or second quadrant of the complex plane (i.e., Imzi 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, ivalidi can only return as ivalidi=0.
If however, Imzi is less than zero, accuracy may be lost in two ways; firstly, in the evaluation of e-zi2, if Im-zi2 is large, in which case a warning will be issued through ivalidi=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 ivalidi=1, 2 or 3.

8 Parallelism and Performance

s15drf is not threaded in any implementation.

9 Further Comments

The time taken for a call of s15drf depends on the argument zi, the time increasing as zi0.0.
s15drf may be used to compute values of erfczi and erfzi for complex zi by the relations erfczi=e-zi2wizi, erfzi=1-erfczi. (For real arguments, s15arf and s15asf should be used.)

10 Example

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

10.1 Program Text

Program Text (s15drfe.f90)

10.2 Program Data

Program Data (s15drfe.d)

10.3 Program Results

Program Results (s15drfe.r)