NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ac: FL CL CPP AD

NAG FL Interface
s15acf (compcdf_normal)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ac: FL CL CPP AD

▸▿ 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. 2021

1 Purpose

s15acf returns the value of the complement of the cumulative Normal distribution function,

Q (x)

, via the function name.

2 Specification

Fortran Interface

Function s15acf (

Real (Kind=nag_wp)	::	s15acf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nag.h>

double

s15acf_ (const double *x, Integer *ifail)

The routine may be called by the names s15acf or nagf_specfun_compcdf_normal.

3 Description

s15acf 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 routine is based on the fact that

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

and it calls s15adf to obtain the necessary value of

erfc

, the complementary error function.

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, 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

1

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

s15acf 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 (s15acfe.f90)

10.2 Program Data

Program Data (s15acfe.d)

10.3 Program Results

Program Results (s15acfe.r)

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s15ac: FL CL CPP AD

© The Numerical Algorithms Group Ltd. 2021