NAG Library Function Document

nag_kelvin_ker_vector (s19aqc)

▸▿ 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_ker_vector (s19aqc) returns an array of values for the Kelvin function

\ker x

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_kelvin_ker_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3 Description

nag_kelvin_ker_vector (s19aqc) 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 function 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 function fails.

4 References

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

5 Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x [n]$ – const doubleInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x [i - 1] > 0.0

, for

i = 1, 2, \dots, n

3: $f [n]$ – doubleOutput

On exit:

\ker x_{i}

, the function values.

4: $ivalid [n]$ – IntegerOutput

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i}$ is too large, the result underflows. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_GT in nag_kelvin_ker (s19acc), as defined in the Users' Note for your implementation.
$ivalid [i - 1] = 2$: $x_{i} \leq 0.0$ , the function is undefined. $f [i - 1]$ contains $0.0$ .

5: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NW_IVALID: On entry, at least one value of x was invalid.
Check ivalid for more 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 function returns zero. Note that for large

x

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

8 Parallelism and Performance

Not applicable.

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 =

NW_IVALID.

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 Function Documentnag_kelvin_ker_vector (s19aqc)

▸▿ 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 Library Function Document

nag_kelvin_ker_vector (s19aqc)