Interfaces: FL CL AD

s19ac: FL CL AD

NAG FL Interface
s19acf (kelvin_ker)

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NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s19ac: FL CL AD

▸▿ 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

s19acf returns a value for the Kelvin function

\ker x

, via the function name.

2 Specification

Fortran Interface

Function s19acf (

x, ifail)

Real (Kind=nag_wp)	::	s19acf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nag.h>

double

s19acf_ (const double *x, Integer *ifail)

The routine may be called by the names s19acf or nagf_specfun_kelvin_ker.

3 Description

s19acf 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 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: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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, $x = 〈value〉$ . The function returns zero.
Constraint: $x \leq 〈value〉$ .
x is too large, the result underflows and the function returns zero.

$ifail = 2$: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .
The function is undefined and returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

s19acf 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 the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s19ac: FL CL AD

NAG FL Interfaces19acf (kelvin_​ker)

▸▿ 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 FL Interface
s19acf (kelvin_ker)