Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	f(n)

C Header Interface

#include <nag.h>

void	s18arf_ (const Integer n, const double x[], double f[], Integer ivalid[], Integer ifail)

The routine may be called by the names s18arf or nagf_specfun_bessel_k1_real_vector.

3 Description

s18arf evaluates an approximation to the modified Bessel function of the second kind

K_{1} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

K_{1} (x)

is undefined for

x \leq 0

and the routine will fail for such arguments.

The routine 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

it is impossible to calculate

\frac{1}{x}

without overflow and the routine 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

NIST Digital Library of Mathematical Functions

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x (i) > 0.0

, for

i = 1, 2, \dots, n

3: $f (n)$ – Real (Kind=nag_wp) array Output

On exit:

K_{1} (x_{i})

, the function values.

4: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i} \leq 0.0$ , $K_{1} (x_{i})$ is undefined. $f (i)$ contains $0.0$ .
$ivalid (i) = 2$: $x_{i}$ is too small, there is a danger of overflow. $f (i)$ contains zero. The threshold value is the same as for $ifail = 2$ in s18adf, as defined in the Users' Note for your implementation.

5: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

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

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, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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

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 routine 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

s18arf is not threaded in any implementation.

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.

s18ar: FL CL CPP AD

NAG FL Interfaces18arf (bessel_​k1_​real_​vector)

▸▿ 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
s18arf (bessel_k1_real_vector)