NAG FL Interface
s15abf (cdf_​normal)

1 Purpose

s15abf returns the value of the cumulative Normal distribution function, Px, via the function name.

2 Specification

Fortran Interface
Function s15abf ( x, ifail)
Real (Kind=nag_wp) :: s15abf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s15abf_ (const double *x, Integer *ifail)
The routine may be called by the names s15abf or nagf_specfun_cdf_normal.

3 Description

s15abf evaluates an approximate value for the cumulative Normal distribution function
Px = 12π -x e-u2/2 du .  
The routine is based on the fact that
Px = 12 erfc-x2  
and it calls s15adf to obtain a value of erfc for the appropriate argument.

4 References

NIST Digital Library of Mathematical Functions

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. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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 this routine. The argument ifail is included for consistency with other routines in this chapter.

7 Accuracy

Because of its close relationship with erfc, the accuracy of this routine is very similar to that in s15adf. If ε and δ are the relative errors in result and argument, respectively, they are in principle related by
ε x e -12 x2 2πPx δ  
so that the relative error in the argument, x, is amplified by a factor, x e -12 x2 2πPx , in the result.
For x small and for x positive this factor is always less than 1 and accuracy is mainly limited by machine precision.
For large negative x the factor behaves like x2 and hence to a certain extent relative accuracy is unavoidably lost.
However, the absolute error in the result, E, is given by
E x e -12 x2 2π δ  
so absolute accuracy can be guaranteed for all x.

8 Parallelism and Performance

s15abf 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 (s15abfe.f90)

10.2 Program Data

Program Data (s15abfe.d)

10.3 Program Results

Program Results (s15abfe.r)