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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_compcdf_normal (s15ac)

## Purpose

nag_specfun_compcdf_normal (s15ac) returns the value of the complement of the cumulative Normal distribution function, Q(x)$Q\left(x\right)$, via the function name.

## Syntax

[result, ifail] = s15ac(x)
[result, ifail] = nag_specfun_compcdf_normal(x)

## Description

nag_specfun_compcdf_normal (s15ac) evaluates an approximate value for the complement of the cumulative Normal distribution function
 ∞ Q(x) = 1/(sqrt(2π)) ∫ e − u2 / 2du. x
$Q(x)=12π∫x∞e-u2/2du.$
The function is based on the fact that
 Q(x) = (1/2)erfc(x/(sqrt(2))) $Q(x)=12erfc(x2)$
and it calls nag_specfun_erfc_real (s15ad) to obtain the necessary value of erfc$\mathit{erfc}$, the complementary error function.

## References

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

## Parameters

### Compulsory Input Parameters

1:     x – double scalar
The argument x$x$ of the function.

None.

None.

### Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

There are no failure exits from this function. The parameter ifail is included for consistency with other functions in this chapter.

## Accuracy

Because of its close relationship with erfc$\mathit{erfc}$ the accuracy of this function is very similar to that in nag_specfun_erfc_real (s15ad). If ε$\epsilon$ and δ$\delta$ are the relative errors in result and argument, respectively, then in principle they are related by
 |ε| ≃ |( x e − x2 / 2 )/(sqrt(2π)Q(x))δ| . $|ε|≃ | x e -x2/2 2πQ(x) δ | .$
For x$x$ negative or small positive this factor is always less than one and accuracy is mainly limited by machine precision. For large positive x$x$ we find εx2δ$\epsilon \sim {x}^{2}\delta$ and hence to a certain extent relative accuracy is unavoidably lost. However the absolute error in the result, E$E$, is given by
 |E| ≃ |( x e − x2 / 2 )/(sqrt(2π))δ| $|E|≃ | x e -x2/2 2π δ |$
and since this factor is always less than one absolute accuracy can be guaranteed for all x$x$.

None.

## Example

```function nag_specfun_compcdf_normal_example
x = -20;
[result, ifail] = nag_specfun_compcdf_normal(x)
```
```

result =

1

ifail =

0

```
```function s15ac_example
x = -20;
[result, ifail] = s15ac(x)
```
```

result =

1

ifail =

0

```