NAG Library Routine Document

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nagmk26.h>

void	s19aqf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

3

Description

s19aqf evaluates an approximation to the Kelvin function

\ker x_{i}

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note: for

x < 0

the function is undefined and at

x = 0

it is infinite so we need only consider

x > 0

The routine is based on several Chebyshev expansions:

For

0 < x \leq 1

\ker x = - f (t) \log (x) + \frac{π}{16} x^{2} g (t) + y (t)

where

f (t)

g (t)

and

y (t)

are expansions in the variable

t = 2 x^{4} - 1

For

1 < x \leq 3

\ker x = \exp (- \frac{11}{16} x) q (t)

where

q (t)

is an expansion in the variable

t = x - 2

For

x > 3

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

where

β = \frac{x}{\sqrt{2}} + \frac{π}{8}

, and

c (t)

and

d (t)

are expansions in the variable

t = \frac{6}{x} - 1

When

x

is sufficiently close to zero, the result is computed as

\ker x = - γ - \log (\frac{x}{2}) + (π - \frac{3}{8} x^{2}) \frac{x^{2}}{16}

and when

x

is even closer to zero, simply as

\ker x = - γ - \log (\frac{x}{2})

For large

x

\ker x

is asymptotically given by

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

and this becomes so small that it cannot be computed without underflow and the routine fails.

4

References

NIST Digital Library of Mathematical Functions

5

Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) arrayInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x (i) > 0.0

, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) arrayOutput

On exit:

\ker x_{i}

, the function values.

4: $ivalid (n)$ – Integer arrayOutput

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i}$ is too large, the result underflows. $f (i)$ contains zero. The threshold value is the same as for $ifail = 1$ in s19acf, as defined in the Users' Note for your implementation.
$ivalid (i) = 2$: $x_{i} \leq 0.0$ , the function is undefined. $f (i)$ contains $0.0$ .

5: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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 routine returns zero. Note that for large

x

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

8

Parallelism and Performance

s19aqf 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

ifail = 1

10

Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

NAG Library Routine Document

s19aqf (kelvin_ker_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

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