NAG FL Interface
s15aqf (compcdf_​normal_​vector)

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1 Purpose

s15aqf returns an array of values of the complement of the cumulative Normal distribution function, Q(x).

2 Specification

Fortran Interface
Subroutine s15aqf ( n, x, f, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s15aqf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s15aqf or nagf_specfun_compcdf_normal_vector.

3 Description

s15aqf evaluates approximate values for the complement of the cumulative Normal distribution function
Q(x) = 12π x e-u2/2 du ,  
for an array of arguments xi, for i=1,2,,n.
The routine is based on the fact that
Q(x) = 12 erfc(x2)  
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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x(n) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: Q(xi), the function values.
4: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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| | x e -x2/2 2πQ(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 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.

8 Parallelism and Performance

s15aqf is not threaded in any implementation.

9 Further Comments


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 (s15aqfe.f90)

10.2 Program Data

Program Data (s15aqfe.d)

10.3 Program Results

Program Results (s15aqfe.r)