NAG Library Routine Document

S15ABF

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 parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 parameter IFAIL is included for consistency with other routines in this chapter.

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

None.

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.