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NAG Toolbox: nag_specfun_compcdf_normal (s15ac)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_compcdf_normal (s15ac) returns the value of the complement of the cumulative Normal distribution function, Qx, via the function name.

Syntax

[result, ifail] = s15ac(x)
[result, ifail] = nag_specfun_compcdf_normal(x)

Description

nag_specfun_compcdf_normal (s15ac) evaluates an approximate value for the complement of the cumulative Normal distribution function
Qx=12πxe-u2/2du.  
The function is based on the fact that
Qx=12erfcx2  
and it calls nag_specfun_erfc_real (s15ad) to obtain the necessary value of erfc, the complementary error function.

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 x of the function.

Optional Input Parameters

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=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 argument ifail is included for consistency with other functions in this chapter.

Accuracy

Because of its close relationship with erfc the accuracy of this function is very similar to that in nag_specfun_erfc_real (s15ad). If ε and δ are the relative errors in result and argument, respectively, then in principle they are related by
ε x e -x2/2 2πQx δ .  
For x negative or small positive this factor is always less than one and accuracy is mainly limited by machine precision. For large positive x we find εx2δ and hence to a certain extent relative accuracy is unavoidably lost. However the absolute error in the result, E, is given by
E x e -x2/2 2π δ  
and since this factor is always less than one absolute accuracy can be guaranteed for all x.

Further Comments

None.

Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s15ac_example


fprintf('s15ac example results\n\n');

x = [-20   -1     0    1    2    20];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s15ac(x(j));
end

disp('      x          Q(x)');
fprintf('%12.3e%12.3e\n',[x; result]);


s15ac example results

      x          Q(x)
  -2.000e+01   1.000e+00
  -1.000e+00   8.413e-01
   0.000e+00   5.000e-01
   1.000e+00   1.587e-01
   2.000e+00   2.275e-02
   2.000e+01   2.754e-89

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