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) .  
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

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


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)