NAG Library Function Document

nag_kelvin_ber_vector (s19anc)

▸▿ 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

nag_kelvin_ber_vector (s19anc) returns an array of values for the Kelvin function

ber x

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_kelvin_ber_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3 Description

nag_kelvin_ber_vector (s19anc) evaluates an approximation to the Kelvin function

ber x_{i}

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

ber (- x) = ber x

, so the approximation need only consider

x \geq 0.0

The function is based on several Chebyshev expansions:

For

0 \leq x \leq 5

ber x = \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{5})}^{4} - 1 .

For

x > 5

\begin{array}{l} ber x = & \frac{e^{x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} a (t)) \cos α + \frac{1}{x} b (t) \sin α] \\ + \frac{e^{- x / \sqrt{2}}}{\sqrt{2 π x}} [(1 + \frac{1}{x} c (t)) \sin β + \frac{1}{x} d (t) \cos β], \end{array}

where

α = \frac{x}{\sqrt{2}} - \frac{π}{8}

β = \frac{x}{\sqrt{2}} + \frac{π}{8}

and

a (t)

b (t)

c (t)

, and

d (t)

are expansions in the variable

t = \frac{10}{x} - 1

When

x

is sufficiently close to zero, the result is set directly to

ber 0 = 1.0

For large

x

, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the function must fail.

4 References

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

5 Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x [n]$ – const doubleInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f [n]$ – doubleOutput

On exit:

ber x_{i}

, the function values.

4: $ivalid [n]$ – IntegerOutput

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $abs (x_{i})$ is too large for an accurate result to be returned. $f [i - 1]$ contains zero. The threshold value is the same as for $fail . code =$ NE_REAL_ARG_GT in nag_kelvin_ber (s19aac), as defined in the Users' Note for your implementation.

5: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NW_IVALID: On entry, at least one value of x was invalid.
Check ivalid for more information.

7 Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. 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 precision, then we have:

E ≃ |\frac{x}{\sqrt{2}} ({ber}_{1} x + {bei}_{1} x)| δ

(provided

E

is within machine bounds).

For small

x

the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.

For medium and large

x

, the error behaviour is oscillatory and its amplitude grows like

\sqrt{\frac{x}{2 π}} e^{x / \sqrt{2}}

. Therefore it is not possible to calculate the function with any accuracy when

\sqrt{x} e^{x / \sqrt{2}} > \frac{\sqrt{2 π}}{δ}

. Note that this value of

x

is much smaller than the minimum value of

x

for which the function overflows.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

NAG Library Function Documentnag_kelvin_ber_vector (s19anc)