NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ab: FL CL CPP AD PY MB

NAG FL Interface
s15abf (cdf_normal)

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NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ab: FL CL CPP AD PY MB

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2022

1 Purpose

s15abf returns the value of the cumulative Normal distribution function,

P (x)

, via the function name.

2 Specification

Fortran Interface

Function s15abf (

Real (Kind=nag_wp)	::	s15abf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nag.h>

double

s15abf_ (const double *x, Integer *ifail)

The routine may be called by the names s15abf or nagf_specfun_cdf_normal.

3 Description

s15abf 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 routine is based on the fact that

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

and it calls s15adf to obtain a value of

erfc

for the appropriate argument.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.
2: $ifail$ – Integer Input/Output: There are no failure exits from this routine. The argument ifail is included for consistency with other routines in this chapter.

6 Error Indicators and Warnings

None.

7 Accuracy

Because of its close relationship with

erfc

, the accuracy of this routine is very similar to that in s15adf. 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

Background information to multithreading can be found in the Multithreading documentation.

s15abf 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.

10.1 Program Text

Program Text (s15abfe.f90)

10.2 Program Data

Program Data (s15abfe.d)

10.3 Program Results

Program Results (s15abfe.r)

NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ab: FL CL CPP AD PY MB

© The Numerical Algorithms Group Ltd. 2022