# NAG FL Interfaces15apf (cdf_​normal_​vector)

## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine s15apf ( n, x, f,
 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.

## 3Description

s15apf evaluates approximate values of the cumulative Normal distribution function
 $Px = 12π ∫-∞x e-u2/2 du ,$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
The routine is based on the fact that
 $Px = 12 erfc-x2$
and it calls s15adf to obtain a value of $\mathit{erfc}$ for the appropriate argument.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $P\left({x}_{i}\right)$, the function values.
4: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . 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 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 is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

Because of its close relationship with $\mathit{erfc}$, the accuracy of this routine is very similar to that in s15adf. If $\epsilon$ and $\delta$ 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, $\frac{x{e}^{-\frac{1}{2}{x}^{2}}}{\sqrt{2\pi }P\left(x\right)}$, 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 $\text{}\sim {x}^{2}$ 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 $x$.

## 8Parallelism and Performance

s15apf is not threaded in any implementation.

None.

## 10Example

This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.

### 10.1Program Text

Program Text (s15apfe.f90)

### 10.2Program Data

Program Data (s15apfe.d)

### 10.3Program Results

Program Results (s15apfe.r)