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NAG Library Function Document

nag_kelvin_ber (s19aac)

▸▿ 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

1 Purpose

nag_kelvin_ber (s19aac) returns a value for the Kelvin function

ber x

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_kelvin_ber (double x, NagError *fail)

3 Description

nag_kelvin_ber (s19aac) evaluates an approximation to the Kelvin function

ber x

Note:

ber (- x) = ber x

, so the approximation need only consider

x \geq 0.0

The function is based on several Chebyshev expansions:

For

0 \leq x \leq 5

ber x = \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{5})}^{4} - 1 .

For

x > 5

\begin{array}{l} ber x = & \frac{e^{x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} a (t)) \cos α + \frac{1}{x} b (t) \sin α] \\ + \frac{e^{- x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} c (t)) \sin β + \frac{1}{x} d (t) \cos β], \end{array}

where

α = \frac{x}{\sqrt{2}} - \frac{π}{8}

β = \frac{x}{\sqrt{2}} + \frac{π}{8}

and

a (t)

b (t)

c (t)

, and

d (t)

are expansions in the variable

t = \frac{10}{x} - 1

When

x

is sufficiently close to zero, the result is set directly to

ber 0 = 1.0

For large

x

, 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

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.
2: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT: On entry, $x = 〈value〉$ .
Constraint: $|x| \leq 〈value〉$ .
$|x|$ is too large for an accurate result to be returned and the function returns zero.

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 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}} ({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 not possible to calculate the function 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

nag_kelvin_ber (s19aac) 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.

NAG Library Function Documentnag_kelvin_ber (s19aac)