1: $x$ – double Input: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $fail$ – 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_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.
NE_REAL_ARG_GT: On entry, $x = ⟨ value ⟩$ . The function returns zero.
Constraint: $x \leq ⟨ value ⟩$ .
x is too large, the result underflows and the function returns zero.
NE_REAL_ARG_LE: On entry, $x = ⟨ value ⟩$ .
Constraint: $x > 0.0$ .
The function is undefined and returns zero.

7 Accuracy

Let

E

be the absolute error in the result,

ε

be the relative error in the result 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}} (\ker_{1} x + {kei}_{1} x) | δ,

ε ≃ | \frac{x}{\sqrt{2}} \frac{\ker_{1} x + {kei}_{1} x}{\ker x} | δ .

For very small

x

, the relative error amplification factor is approximately given by

\frac{1}{| \log (x) |}

, which implies a strong attenuation of relative error. However,

ε

in general cannot be less than the machine precision.

For small

x

, errors are damped by the function and hence are limited by the machine precision.

For medium and large

x

, the error behaviour, like the function itself, is oscillatory, and hence only the absolute accuracy for the function can be maintained. For this range of

x

, the amplitude of the absolute error decays like

\sqrt{\frac{π x}{2}} e^{- x / \sqrt{2}}

which implies a strong attenuation of error. Eventually,

\ker x

, which asymptotically behaves like

\sqrt{\frac{π}{2 x}} e^{- x / \sqrt{2}}

, becomes so small that it cannot be calculated without causing underflow, and the function returns zero. Note that for large

x

the errors are dominated by those of the standard math library function exp.

8 Parallelism and Performance

s19acc is not threaded in any implementation.

9 Further Comments

Underflow may occur for a few values of

x

close to the zeros of

\ker x

, below the limit which causes a failure with

fail . code =

NE_REAL_ARG_GT.

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.

s19ac: FL CL CPP AD

NAG CL Interfaces19acc (kelvin_​ker)

▸▿ 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
s19acc (kelvin_ker)