s15ab: FL CL CPP AD

NAG CL Interface
s15abc (cdf_normal)

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NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ab: FL CL CPP AD

▸▿ 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

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Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

s15abc returns the value of the cumulative Normal distribution function,

P (x)

2 Specification

#include <nag.h>

double

s15abc (double x)

The function may be called by the names: s15abc, nag_specfun_cdf_normal or nag_cumul_normal.

3 Description

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 s15adc to obtain a value of

erfc

for the appropriate argument.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: $x$ – double Input: 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 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

1

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

s15abc is not threaded in any implementation.

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.

s15ab: FL CL CPP AD

NAG CL Interfaces15abc (cdf_​normal)

▸▿ 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

NAG CL Interface
s15abc (cdf_normal)