1: $n$ – Integer Input: On entry: $n$ , the number of points.

Constraint: $n \geq 0$ .
2: $x (n)$ – Real (Kind=nag_wp) array Input: On entry: the argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .
3: $f (n)$ – Real (Kind=nag_wp) array Output: On exit: $P (x_{i})$ , 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

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: 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, 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

1

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

s15apf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

s15ap: FL CL CPP AD

NAG FL Interfaces15apf (cdf_​normal_​vector)

▸▿ 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

NAG FL Interface
s15apf (cdf_normal_vector)