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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_kelvin_ker (s19ac)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_kelvin_ker (s19ac) returns a value for the Kelvin function

\ker x

, via the function name.

Syntax

[result, ifail] = s19ac(x)

[result, ifail] = nag_specfun_kelvin_ker(x)

Description

nag_specfun_kelvin_ker (s19ac) evaluates an approximation to the Kelvin function

\ker x

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.

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Constraint: $x > 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, x is too large: the result underflows. On soft failure, the function returns zero.

$ifail = 2$: On entry, $x \leq 0.0$ : the function is undefined. On soft failure the function returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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.

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

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s19ac_example

function s19ac_example


fprintf('s19ac example results\n\n');

x = [0.1   1    2.5   5   10   15];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s19ac(x(j));
end

disp('      x          ker(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s19ac example results

      x          ker(x)
   1.000e-01   2.420e+00
   1.000e+00   2.867e-01
   2.500e+00  -6.969e-02
   5.000e+00  -1.151e-02
   1.000e+01   1.295e-04
   1.500e+01  -1.514e-08

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Chapter Contents

Chapter Introduction

NAG Toolbox