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NAG Toolbox: nag_specfun_cdf_normal (s15ab)

Purpose

nag_specfun_cdf_normal (s15ab) returns the value of the cumulative Normal distribution function, P(x)P(x), via the function name.

Syntax

[result, ifail] = s15ab(x)
[result, ifail] = nag_specfun_cdf_normal(x)

Description

nag_specfun_cdf_normal (s15ab) evaluates an approximate value for the cumulative Normal distribution function
x
P(x) = 1/(sqrt(2π))eu2 / 2du.
P(x)=12π-xe-u2/2du.
The function is based on the fact that
P(x) = (1/2)erfc((x)/(sqrt(2)))
P(x)=12erfc(-x2)
and it calls nag_specfun_erfc_real (s15ad) to obtain a value of erfcerfc for the appropriate argument.

References

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

Parameters

Compulsory Input Parameters

1:     x – double scalar
The argument xx of the function.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

There are no failure exits from this function. The parameter ifail is included for consistency with other functions in this chapter.

Accuracy

Because of its close relationship with erfcerfc, the accuracy of this function is very similar to that in nag_specfun_erfc_real (s15ad). If εε and δδ are the relative errors in result and argument, respectively, they are in principle related by
|ε| |( x e (1/2) x2 )/(sqrt(2π)P(x))δ|
|ε| | x e -12 x2 2πP(x) δ |
so that the relative error in the argument, xx, is amplified by a factor, (xe(1/2)x2)/(sqrt(2π)P(x)) xe-12x2 2πP(x) , in the result.
For xx small and for xx positive this factor is always less than one and accuracy is mainly limited by machine precision.
For large negative xx the factor behaves like x2x2 and hence to a certain extent relative accuracy is unavoidably lost.
However the absolute error in the result, EE, is given by
|E| |( x e (1/2) x2 )/(sqrt(2π))δ|
|E| | x e -12 x2 2π δ |
so absolute accuracy can be guaranteed for all xx.

Further Comments

None.

Example

function nag_specfun_cdf_normal_example
x = -20;
[result, ifail] = nag_specfun_cdf_normal(x)
 

result =

   2.7536e-89


ifail =

                    0


function s15ab_example
x = -20;
[result, ifail] = s15ab(x)
 

result =

   2.7536e-89


ifail =

                    0



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Chapter Contents
Chapter Introduction
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