, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner;, therefore, the routine must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.
2: $ifail$ – 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, $x = ⟨ value ⟩$ .
Constraint: $| x | \leq ⟨ value ⟩$ .
$| x |$ is too large for an accurate result to be returned and the function returns zero.

$ifail = - 99$: 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

Since the function is oscillatory, the absolute error rather than the relative error is important. Let

E

be the absolute error in the function, and

δ

be the relative error in the argument. If

δ

is somewhat larger than the machine precision, then we have:

E ≃ | \frac{x}{\sqrt{2}} (- {ber}_{1} x + {bei}_{1} x) | δ

(provided

E

is within machine bounds).

For small

x

the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.

For medium and large

x

, the error behaviour is oscillatory and its amplitude grows like

\sqrt{\frac{x}{2 π}} e^{x / \sqrt{2}}

. Therefore, it is impossible to calculate the functions with any accuracy when

\sqrt{x} e^{x / \sqrt{2}} > \frac{\sqrt{2 π}}{δ}

. Note that this value of

x

is much smaller than the minimum value of

x

for which the function overflows.

8 Parallelism and Performance

s19abf is not threaded in any implementation.

9 Further Comments

None.

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.

s19ab: FL CL CPP AD

NAG FL Interfaces19abf (kelvin_​bei)

▸▿ 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
s19abf (kelvin_bei)