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) .  
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

s15aef is not threaded in any implementation.

9 Further Comments

None.

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)