NAG Library Function Document

nag_kelvin_bei (s19abc)

+− 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_bei (s19abc) returns a value for the Kelvin function

bei x

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_kelvin_bei (double x, NagError *fail)

3 Description

nag_kelvin_bei (s19abc) evaluates an approximation to the Kelvin function

bei x

Note:

bei (- x) = bei 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

bei x = \frac{x^{2}}{4} \sum_{r = 0}^{'} a_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{5})}^{4} - 1;

For

x > 5

bei x = \frac{e^{x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} a (t)) \sin α - \frac{1}{x} b (t) \cos α]

+ \frac{e^{x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} c (t)) \cos β - \frac{1}{x} d (t) \sin β]

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 computed as

bei x = \frac{x^{2}}{4}

. If this result would underflow, the result returned is

bei x = 0.0

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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 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

Not applicable.

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_bei (s19abc)