Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

```