NAG Library ManualKeyword Search:

NAG Library Routine Document

s15abf (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

1

Purpose

s15abf returns the value of the cumulative Normal distribution function,

P (x)

, via the function name.

2

Specification

Fortran Interface

Function s15abf (

x, ifail)

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

C Header Interface

#include nagmk26.h

double

s15abf_ ( const double *x, Integer *ifail)

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

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

5

Arguments

1: $x$ – Real (Kind=nag_wp)Input: On entry: the argument $x$ of the function.
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

There are no failure exits from this routine. The argument ifail is included for consistency with other routines in this chapter.

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 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

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.

NAG Library Routine Document

s15abf (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