NAG FL Interface
s15aef (erf_​real)

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1 Purpose

s15aef returns the value of the error function erf(x), via the function name.

2 Specification

Fortran Interface
Function s15aef ( x, ifail)
Real (Kind=nag_wp) :: s15aef
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s15aef_ (const double *x, Integer *ifail)
The routine may be called by the names s15aef or nagf_specfun_erf_real.

3 Description

s15aef calculates an approximate value for the error function
erf(x) = 2π 0x e-t2 dt = 1-erfc(x) .  
Unless stated otherwise in the Users' Note, s15aef calls the 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 erf(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 erf(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, s15aef returns erf(x)=1; for x-xhi it returns erf(x)=-1.

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 s15aef. The argument ifail has been included for consistency with other routines in this chapter.

7 Accuracy

See Section 7 in s15adf.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s15aef 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 (s15aefe.f90)

10.2 Program Data

Program Data (s15aefe.d)

10.3 Program Results

Program Results (s15aefe.r)