NAG FL Interface
s15apf (cdf_​normal_​vector)

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1 Purpose

s15apf returns an array of values of the cumulative Normal distribution function, P(x).

2 Specification

Fortran Interface
Subroutine s15apf ( n, x, f, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s15apf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s15apf or nagf_specfun_cdf_normal_vector.

3 Description

s15apf evaluates approximate values of the cumulative Normal distribution function
P(x) = 12π -x e-u2/2 du ,  
for an array of arguments xi, for i=1,2,,n.
The routine is based on the fact that
P(x) = 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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x(n) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: P(xi), the function values.
4: 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

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 2πP(x) δ|  
so that the relative error in the argument, x, is amplified by a factor, x e -12 x2 2πP(x) , 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

s15apf is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s15apfe.f90)

10.2 Program Data

Program Data (s15apfe.d)

10.3 Program Results

Program Results (s15apfe.r)