NAG Library Routine Document

S19ADF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S19ADF returns a value for the Kelvin function

kei x

via the function name.

2 Specification

FUNCTION S19ADF (

X, IFAIL)

REAL (KIND=nag_wp) S19ADF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S19ADF 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 routine 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 routine 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 routine fails.

4 References

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

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the function.
Constraint: $X \geq 0.0$ .
2: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, X is too large: the result underflows. On soft failure, the routine returns zero. See also the Users' Note for your implementation.

$IFAIL = 2$: On entry, $X < 0.0$ : the function is undefined. On soft failure the routine returns zero.

7 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 routine returns zero. Note that for large

x

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

8 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

9 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 Routine DocumentS19ADF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S19ADF