NAG Library Routine Document

s15abf (cdf_normal)


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


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 <nagmk26.h>
double  s15abf_ (const double *x, Integer *ifail)


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.


NIST Digital Library of Mathematical Functions


1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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).

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.


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.

Parallelism and Performance

s15abf is not threaded in any implementation.

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

Program Text

Program Text (s15abfe.f90)

Program Data

Program Data (s15abfe.d)

Program Results

Program Results (s15abfe.r)