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

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x [n]$ – const double Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f [n]$ – double Output

On exit:

bei x_{i}

, the function values.

4: $ivalid [n]$ – Integer Output

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $abs (x_{i})$ is too large for an accurate result to be returned. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_GT in s19abc , as defined in the Users' Note for your implementation.

5: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of x was invalid.
Check ivalid for more 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

s19apc 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.

s19ap: FL CL CPP AD

NAG CL Interfaces19apc (kelvin_​bei_​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 CL Interface
s19apc (kelvin_bei_vector)