NAG CL Interface
s15adc (erfc_​real)

Internal changes have been made to this routine in some implementations at Mark 27.1.1.
This document reflects the updated function. The documentation of the Mark 27.1(.0) implementation is also available here.
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1 Purpose

s15adc returns the value of the complementary error function, erfc(x).

2 Specification

#include <nag.h>
double  s15adc (double x)
The function may be called by the names: s15adc, nag_specfun_erfc_real or nag_erfc.

3 Description

s15adc calculates an approximate value for the complement of the error function
erfc(x) = 2π x e-t2 dt = 1-erf(x) .  
Unless stated otherwise in the Users' Note, s15adc calls the complementary error function supplied by the compiler used for your implementation; as such, details of the underlying algorithm should be obtained from the documentation supplied by the compiler vendor. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
Let x^ be the root of the equation erfc(x)-erf(x)=0 (then x^0.46875). For |x|x^ the value of erfc(x) is based on the following rational Chebyshev expansion for erf(x):
erf(x) 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 erfc(x) is based on a rational Chebyshev expansion for erfc(x): for x^<|x|4 the value is based on the expansion
erfc(x) ex2 R,m (x) ;  
and for |x|>4 it is based on the expansion
erfc(x) ex2 x (1π+1x2R,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 X02BEC).
For |x|xhi there is a danger of setting underflow in erfc(x) (the value of xhi is given in the Users' Note for your implementation).. For xxhi, s15adc returns erfc(x)=0; for x-xhi it returns erfc(x)=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: x double Input
On entry: the argument x of the function.

6 Error Indicators and Warnings

None.

7 Accuracy

Unless stated otherwise in the Users' Note, s15adc calls the complementary error function supplied by the compiler used for your implementation. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
If δ and ε are relative errors in the argument and result, respectively, then in principle
|ε| | 2x e -x2 πerfc(x) δ| .  
That is, the relative error in the argument, x, is amplified by a factor 2xe-x2 πerfc(x) 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

s15adc is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this function as follows:
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

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

10.1 Program Text

Program Text (s15adce.c)

10.2 Program Data

Program Data (s15adce.d)

10.3 Program Results

Program Results (s15adce.r)