NAG Library Function Document

nag_cumul_normal (s15abc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_cumul_normal (s15abc) returns the value of the cumulative Normal distribution function,

P (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_cumul_normal (double x)

3 Description

nag_cumul_normal (s15abc) evaluates an approximate value for the cumulative Normal distribution function

P (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} e^{- u^{2} / 2} d u .

The function is based on the fact that

P (x) = \frac{1}{2} erfc (\frac{- x}{\sqrt{2}})

and it calls nag_erfc (s15adc) to obtain a value of

erfc

for the appropriate argument.

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.

6 Error Indicators and Warnings

None.

7 Accuracy

Because of its close relationship with

erfc

, the accuracy of this function is very similar to that in nag_erfc (s15adc). If

ε

and

δ

are the relative errors in result and argument, respectively, they are in principle related by

|ε| ≃ |\frac{x e^{- \frac{1}{2} x^{2}}}{\sqrt{2 π} P (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 π} P (x)}

, in the result.

For

x

small and for

x

positive this factor is always less than one and accuracy is mainly limited by machine precision.

For large negative

x

the factor behaves like

\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| ≃ |\frac{x e^{- \frac{1}{2} x^{2}}}{\sqrt{2 π}} δ|

so absolute accuracy can be guaranteed for all

x

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_cumul_normal (s15abc)