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NAG Library Function Document

nag_bessel_k1 (s18adc)

▸▿ 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_bessel_k1 (s18adc) returns the value of the modified Bessel function

K_{1} (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_bessel_k1 (double x, NagError *fail)

3 Description

nag_bessel_k1 (s18adc) evaluates an approximation to the modified Bessel function of the second kind

K_{1} (x)

Note:

K_{1} (x)

is undefined for

x \leq 0

and the function will fail for such arguments.

The function is based on five Chebyshev expansions:

For

0 < x \leq 1

K_{1} (x) = \frac{1}{x} + x \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) - x \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = 2 x^{2} - 1 .

For

1 < x \leq 2

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 x - 3 .

For

2 < x \leq 4

K_{1} (x) = e^{- x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t),   where ​ t = x - 3 .

For

x > 4

K_{1} (x) = \frac{e^{- x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} e_{r} T_{r} (t),   where ​ t = \frac{9 - x}{1 + x} .

For

x

near zero,

K_{1} (x) ≃ \frac{1}{x}

. This approximation is used when

x

is sufficiently small for the result to be correct to machine precision. For very small

x

on some machines, it is impossible to calculate

\frac{1}{x}

without overflow and the function must fail.

For large

x

, where there is a danger of underflow due to the smallness of

K_{1}

, the result is set exactly to zero.

4 References

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

5 Arguments

1: $x$ – doubleInput: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LE: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .
$K_{0}$ is undefined and the function returns zero.
NE_REAL_ARG_TOO_SMALL: On entry, $x = 〈value〉$ .
Constraint: $x > 〈value〉$ .
x is too small, there is a danger of overflow and the function returns approximately the largest representable value.

7 Accuracy

Let

δ

and

ε

be the relative errors in the argument and result respectively.

δ

is somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x K_{0} (x) - K_{1} (x)}{K_{1} (x)}| .

However if

δ

is of the same order as the machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

For small

x

ε ≃ δ

and there is no amplification of errors.

For large

x

ε ≃ x δ

and we have strong amplification of the relative error. Eventually

K_{1}

, which is asymptotically given by

\frac{e^{- x}}{\sqrt{x}}

, becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large

x

the errors will be dominated by those of the standard function exp.

Figure 1

8 Parallelism and Performance

nag_bessel_k1 (s18adc) is not threaded in any implementation.

9 Further Comments

None.

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.

NAG Library Function Documentnag_bessel_k1 (s18adc)