NAG Library Function Document

nag_kelvin_ker (s19acc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_kelvin_ker (s19acc) returns a value for the Kelvin function

\ker x

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_kelvin_ker (double x, NagError *fail)

3 Description

nag_kelvin_ker (s19acc) evaluates an approximation to the Kelvin function

\ker x

The function is based on several Chebyshev expansions.

For large

x

\ker x

is so small that it cannot be computed without underflow and the function evaluation fails.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x > 0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_REAL_ARG_GT: On entry, $x = 〈value〉$ .
Constraint: $x \leq 〈value〉$ .
x is too large, the result underflows and the function returns zero.
NE_REAL_ARG_LE: On entry, x must not be less than or equal to 0.0: $x = 〈value〉$ .
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 ≃ |x (\ker_{1} x + {kei}_{1} x) / \sqrt{2}| δ

ε ≃ |x (\ker_{1} x + {kei}_{1} x) / \sqrt{2} \ker x| δ

For very small

x

, the relative error amplification factor is approximately given by

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{π x / 2} e^{- x / \sqrt{2}}

which implies a strong attenuation of error. Eventually,

\ker x

, which asymptotically behaves like

\sqrt{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 math library function exp.

8 Further Comments

Underflow may occur for a few values of

x

close to the zeros of

\ker x

, which causes a failure NE_REAL_ARG_GT.

9 Example

The following program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_kelvin_ker (s19acc)