# NAG Library Function Document

## 1Purpose

nag_cumul_normal_complem (s15acc) returns the value of the complement of the cumulative Normal distribution function, $Q\left(x\right)$.

## 2Specification

 #include #include
 double nag_cumul_normal_complem (double x)

## 3Description

nag_cumul_normal_complem (s15acc) evaluates an approximate value for the complement of the cumulative Normal distribution function
 $Qx=12π∫x∞e-u2/2du.$
The function is based on the fact that
 $Qx=12erfcx2$
and it calls nag_erfc (s15adc) to obtain the necessary value of $\mathit{erfc}$, the complementary error function.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.

None.

## 7Accuracy

Because of its close relationship with $\mathit{erfc}$ the accuracy of this function is very similar to that in nag_erfc (s15adc). If $\epsilon$ and $\delta$ are the relative errors in result and argument, respectively, then in principle they are related by
 $ε≃ x e -x2/2 2πQx δ .$
For $x$ negative or small positive this factor is always less than one and accuracy is mainly limited by machine precision. For large positive $x$ we find $\epsilon \sim {x}^{2}\delta$ and hence to a certain extent relative accuracy is unavoidably lost. However the absolute error in the result, $E$, is given by
 $E≃ x e -x2/2 2π δ$
and since this factor is always less than one absolute accuracy can be guaranteed for all $x$.

## 8Parallelism and Performance

nag_cumul_normal_complem (s15acc) is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s15acce.c)

### 10.2Program Data

Program Data (s15acce.d)

### 10.3Program Results

Program Results (s15acce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017