NAG FL Interface
s15arf (erfc_​real_​vector)

1 Purpose

s15arf returns an array of values of the complementary error function, erfcx.

2 Specification

Fortran Interface
Subroutine s15arf ( n, x, f, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s15arf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s15arf or nagf_specfun_erfc_real_vector.

3 Description

s15arf calculates approximate values for the complement of the error function
erfcx = 2π x e-t2 dt = 1-erfx ,  
for an array of arguments xi, for i=1,2,,n.
Let x^ be the root of the equation erfcx-erfx=0 (then x^0.46875). For xx^ the value of erfcx is based on the following rational Chebyshev expansion for erfx:
erfx xR,m x2 ,  
where R,m denotes a rational function of degree in the numerator and m in the denominator.
For x>x^ the value of erfcx is based on a rational Chebyshev expansion for erfcx: for x^<x4 the value is based on the expansion
erfcx ex2 R,m x ;  
and for x>4 it is based on the expansion
erfcx ex2 x 1π + 1x2 R,m 1/x2 .  
For each expansion, the specific values of and m are selected to be minimal such that the maximum relative error in the expansion is of the order 10-d, where d is the maximum number of decimal digits that can be accurately represented for the particular implementation (see x02bef).
For xxhi there is a danger of setting underflow in erfcx (the value of xhi is given in the Users' Note for your implementation). For xxhi, s15arf returns erfcx=0; for x-xhi it returns erfcx=2.

4 References

NIST Digital Library of Mathematical Functions
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: xn Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
3: fn Real (Kind=nag_wp) array Output
On exit: erfcxi, the function values.
4: 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:
ifail=1
On entry, n=value.
Constraint: n0.
ifail=-99
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.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If δ and ε are relative errors in the argument and result, respectively, then in principle
ε 2x e -x2 πerfcx δ .  
That is, the relative error in the argument, x, is amplified by a factor 2xe-x2 πerfcx in the result.
The behaviour of this factor is shown in Figure 1.
Figure 1
Figure 1
It should be noted that near x=0 this factor behaves as 2xπ and hence the accuracy is largely determined by the machine precision. Also, for large negative x, where the factor is xe-x2π, accuracy is mainly limited by machine precision. However, for large positive x, the factor becomes 2x2 and to an extent relative accuracy is necessarily lost. The absolute accuracy E is given by
E 2xe-x2π δ  
so absolute accuracy is guaranteed for all x.

8 Parallelism and Performance

s15arf is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (s15arfe.f90)

10.2 Program Data

Program Data (s15arfe.d)

10.3 Program Results

Program Results (s15arfe.r)