# NAG CL Interfaces15apc (cdf_​normal_​vector)

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## 1Purpose

s15apc returns an array of values of the cumulative Normal distribution function, $P\left(x\right)$.

## 2Specification

 #include
 void s15apc (Integer n, const double x[], double f[], NagError *fail)
The function may be called by the names: s15apc, nag_specfun_cdf_normal_vector or nag_cumul_normal_vector.

## 3Description

s15apc evaluates approximate values of the cumulative Normal distribution function
 $P(x) = 12π ∫-∞x e-u2/2 du ,$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
The function is based on the fact that
 $P(x) = 12 erfc(-x2)$
and it calls s15adc 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: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $P\left({x}_{i}\right)$, the function values.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

Because of its close relationship with $\mathit{erfc}$, the accuracy of this function is very similar to that in s15adc. If $\epsilon$ and $\delta$ 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, $\frac{x{e}^{-\frac{1}{2}{x}^{2}}}{\sqrt{2\pi }P\left(x\right)}$, 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

s15apc 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 (s15apce.c)

### 10.2Program Data

Program Data (s15apce.d)

### 10.3Program Results

Program Results (s15apce.r)