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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_kelvin_kei (s19ad)

▸▿ 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_kei (s19ad) returns a value for the Kelvin function

kei x

via the function name.

Syntax

[result, ifail] = s19ad(x)

[result, ifail] = nag_specfun_kelvin_kei(x)

Description

nag_specfun_kelvin_kei (s19ad) evaluates an approximation to the Kelvin function

kei x

Note: for

x < 0

the function is undefined, so we need only consider

x \geq 0

The function is based on several Chebyshev expansions:

For

0 \leq x \leq 1

kei x = - \frac{π}{4} f (t) + \frac{x^{2}}{4} [- g (t) \log (x) + v (t)]

where

f (t)

g (t)

and

v (t)

are expansions in the variable

t = 2 x^{4} - 1

;

For

1 < x \leq 3

kei x = \exp (- \frac{9}{8} x) u (t)

where

u (t)

is an expansion in the variable

t = x - 2

;

For

x > 3

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

For

x < 0

, the function is undefined, and hence the function fails and returns zero.

When

x

is sufficiently close to zero, the result is computed as

kei x = - \frac{π}{4} + (1 - γ - \log (\frac{x}{2})) \frac{x^{2}}{4}

and when

x

is even closer to zero simply as

kei x = - \frac{π}{4} .

For large

x

kei 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: $x \geq 0.0$ .

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, $x < 0.0$ : 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, and

δ

be the relative error in the argument. If

δ

is somewhat larger than the machine representation error, then we have:

E ≃ |\frac{x}{\sqrt{2}} (- \ker_{1} x + {kei}_{1} x)| δ .

For small

x

, errors are attenuated 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 absolute accuracy of 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,

kei x

, which is asymptotically given by

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

,becomes so small that it cannot be calculated without causing underflow and therefore 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

kei 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: s19ad_example

function s19ad_example


fprintf('s19ad 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] = s19ad(x(j));
end

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

s19ad example results

      x          kei(x)
   1.000e-01  -7.769e-01
   1.000e+00  -4.950e-01
   2.500e+00  -1.107e-01
   5.000e+00   1.119e-02
   1.000e+01  -3.075e-04
   1.500e+01   7.963e-06

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox