s19ad: FL CL CPP AD

NAG FL Interface
s19adf (kelvin_kei)

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NAG Library Manual, Mark 27.3

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s19ad: FL CL CPP 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

s19adf returns a value for the Kelvin function

kei x

via the function name.

2 Specification

Fortran Interface

Function s19adf (

x, ifail)

Real (Kind=nag_wp)	::	s19adf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nag.h>

double

s19adf_ (const double *x, Integer *ifail)

The routine may be called by the names s19adf or nagf_specfun_kelvin_kei.

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

NIST Digital Library of Mathematical Functions

5 Arguments

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

Constraint: $x \geq 0.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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: $x \geq 0.0$ .
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, 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 Parallelism and Performance

s19adf is not threaded in any implementation.

9 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

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.

s19ad: FL CL CPP AD

NAG FL Interfaces19adf (kelvin_​kei)

▸▿ 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
s19adf (kelvin_kei)