NAG Library Function Document

nag_cumul_normal_complem (s15acc)

+− 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_complem (s15acc) returns the value of the complement of the cumulative Normal distribution function,

Q (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_cumul_normal_complem (double x)

3 Description

nag_cumul_normal_complem (s15acc) evaluates an approximate value for the complement of the cumulative Normal distribution function

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

The function is based on the fact that

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

and it calls nag_erfc (s15adc) to obtain the necessary value of

erfc

, the complementary error function.

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, then in principle they are related by

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

ε \sim x^{2} δ

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^{- x^{2} / 2}}{\sqrt{2 π}} δ|

and since this factor is always less than one 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_complem (s15acc)