NAG FL Interface
s15adf (erfc_​real)

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1 Purpose

s15adf returns the value of the complementary error function, erfc(x), via the function name.

2 Specification

Fortran Interface
Function s15adf ( x, ifail)
Real (Kind=nag_wp) :: s15adf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s15adf_ (const double *x, Integer *ifail)
The routine may be called by the names s15adf or nagf_specfun_erfc_real.

3 Description

s15adf 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, s15adf 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 x02bef).
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, s15adf 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 Real (Kind=nag_wp) Input
On entry: the argument x 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

There are no failure exits from s15adf. The argument ifail has been included for consistency with other routines in this chapter.

7 Accuracy

Unless stated otherwise in the Users' Note, s15adf 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

s15adf is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this routine 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 (s15adfe.f90)

10.2 Program Data

Program Data (s15adfe.d)

10.3 Program Results

Program Results (s15adfe.r)